[{"data":1,"prerenderedAt":1222},["ShallowReactive",2],{"subject:real-analysis":3,"course-wordcounts":75,"nav:real-analysis":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F04.real-analysis\u002Findex.md","Real Analysis","Calculus rebuilt on proof — the completeness of the reals, metric spaces,\nand the theorems that say exactly when limits, derivatives, and integrals\ndo what calculus assumed they would.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Real analysis rebuilds calculus from the ground up, starting from what the real\nnumbers actually are. The sequence opens with the completeness of the reals — the\nleast-upper-bound property that separates the continuum from the rationals — then\ndevelops sequences, series, and the Cauchy criterion that characterizes\nconvergence intrinsically. From there it moves to the topology of the real line\nand of general metric spaces: open and closed sets, compactness, and\nconnectedness, the abstract home of the theorems that follow. Continuity,\ndifferentiation, and Riemann integration are each given their proper\nepsilon-delta footing, culminating in the mean-value and fundamental theorems.\nA careful treatment of pointwise versus uniform convergence explains exactly when\nlimits, derivatives, and integrals may be interchanged, and the notes close with\nthe calculus of several variables. Organized after Rosenlicht, with Lebl and\nShkoller as the working texts.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,28,32,34,38,40,44,46,48,52,54],{"p":20},"Real analysis is calculus made honest: every limit, derivative, and\nintegral you once took on faith is re-derived from a single axiom about\nthe real numbers, with a proof that says exactly when it holds and when\nit fails.\n",{"fig":22,"n":23,"caption":24,"large":25},"ra-convergence","001","A sequence settles into an \u003Cem>ε\u003C\u002Fem>-band around its limit — eventually\nas close to \u003Cem>L\u003C\u002Fem> as you like.\n",true,{"p":27},"Everything rests on \u003Cstrong>completeness\u003C\u002Fstrong>. The rationals have\nholes; the reals do not, and the one axiom that fills them — every set\nbounded above has a \u003Cem>least\u003C\u002Fem> upper bound — is the engine behind\nevery convergence theorem that follows.\n",{"fig":29,"n":30,"caption":31},"ra-supremum","002","The supremum: the least of all upper bounds of a bounded set.\n",{"p":33},"From completeness comes the language of \u003Cstrong>convergence\u003C\u002Fstrong>.\nA sequence has a limit when its terms are eventually trapped in every\ntolerance you name, and Cauchy's criterion lets you prove one converges\nwithout knowing the limit in advance.\n",{"fig":35,"n":36,"caption":37},"ra-nested","003","Nested closed intervals shrinking to the single point they share.\n",{"p":39},"With limits in hand, \u003Cstrong>continuity\u003C\u002Fstrong> is the promise that a\nfunction's value never jumps away from where its inputs are heading —\nand the theorems that a continuous function on a closed interval attains\nits bounds and hits every value between them.\n",{"fig":41,"n":42,"caption":43},"ra-continuity","004","Continuity as an unbroken graph, against a jump where one-sided\nlimits disagree.\n",{"p":45},"Differentiation and integration follow the same discipline: the\nderivative as a limit of slopes, the Riemann integral as a limit of\nsums, and the mean-value and fundamental theorems that bind them\ntogether with proof.\n",{"p":47},"The subtle turn is interchanging limits. Pointwise convergence preserves\nalmost nothing; \u003Cstrong>uniform\u003C\u002Fstrong> convergence — one \u003Cem>N\u003C\u002Fem> that\nworks for every point at once — is what lets you swap a limit with a\nderivative or an integral.\n",{"fig":49,"n":50,"caption":51},"ra-uniform","005","Uniform convergence: a single \u003Cem>ε\u003C\u002Fem>-tube trapping the whole tail.\n",{"p":53},"The same ideas generalize to \u003Cstrong>metric spaces\u003C\u002Fstrong>, where\ndistance alone defines open sets, compactness, and completeness, and\nthe theorems of the real line reappear in their natural setting.\n",{"p":55},"The reward is judgment: you stop asking whether a calculation looks\nright and start knowing precisely which hypotheses make it true.\n","math","Real analysis re-derives calculus from axioms: what the real numbers are,\nwhat convergence means, and precisely when the operations of calculus are\njustified. The notes build the real line, develop sequences and series,\nmove to metric spaces and topology, and then treat continuity,\ndifferentiation, integration, and the interchange of limits with full\nproofs, ending with the calculus of several variables. Organized after\nRosenlicht, with Lebl and Shkoller as the working texts.\n",false,"md",{},"\u002Freal-analysis",[],"---\ntitle: Real Analysis\nstatus: available\ncategory: math\nblurb: |\n  Calculus rebuilt on proof — the completeness of the reals, metric spaces,\n  and the theorems that say exactly when limits, derivatives, and integrals\n  do what calculus assumed they would.\ndescription: |\n  Real analysis re-derives calculus from axioms: what the real numbers are,\n  what convergence means, and precisely when the operations of calculus are\n  justified. The notes build the real line, develop sequences and series,\n  move to metric spaces and topology, and then treat continuity,\n  differentiation, integration, and the interchange of limits with full\n  proofs, ending with the calculus of several variables. Organized after\n  Rosenlicht, with Lebl and Shkoller as the working texts.\nbrief:\n  - p: |\n      Real analysis is calculus made honest: every limit, derivative, and\n      integral you once took on faith is re-derived from a single axiom about\n      the real numbers, with a proof that says exactly when it holds and when\n      it fails.\n  - fig: ra-convergence\n    n: \"001\"\n    caption: |\n      A sequence settles into an \u003Cem>ε\u003C\u002Fem>-band around its limit — eventually\n      as close to \u003Cem>L\u003C\u002Fem> as you like.\n    large: true\n  - p: |\n      Everything rests on \u003Cstrong>completeness\u003C\u002Fstrong>. The rationals have\n      holes; the reals do not, and the one axiom that fills them — every set\n      bounded above has a \u003Cem>least\u003C\u002Fem> upper bound — is the engine behind\n      every convergence theorem that follows.\n  - fig: ra-supremum\n    n: \"002\"\n    caption: |\n      The supremum: the least of all upper bounds of a bounded set.\n  - p: |\n      From completeness comes the language of \u003Cstrong>convergence\u003C\u002Fstrong>.\n      A sequence has a limit when its terms are eventually trapped in every\n      tolerance you name, and Cauchy's criterion lets you prove one converges\n      without knowing the limit in advance.\n  - fig: ra-nested\n    n: \"003\"\n    caption: |\n      Nested closed intervals shrinking to the single point they share.\n  - p: |\n      With limits in hand, \u003Cstrong>continuity\u003C\u002Fstrong> is the promise that a\n      function's value never jumps away from where its inputs are heading —\n      and the theorems that a continuous function on a closed interval attains\n      its bounds and hits every value between them.\n  - fig: ra-continuity\n    n: \"004\"\n    caption: |\n      Continuity as an unbroken graph, against a jump where one-sided\n      limits disagree.\n  - p: |\n      Differentiation and integration follow the same discipline: the\n      derivative as a limit of slopes, the Riemann integral as a limit of\n      sums, and the mean-value and fundamental theorems that bind them\n      together with proof.\n  - p: |\n      The subtle turn is interchanging limits. Pointwise convergence preserves\n      almost nothing; \u003Cstrong>uniform\u003C\u002Fstrong> convergence — one \u003Cem>N\u003C\u002Fem> that\n      works for every point at once — is what lets you swap a limit with a\n      derivative or an integral.\n  - fig: ra-uniform\n    n: \"005\"\n    caption: |\n      Uniform convergence: a single \u003Cem>ε\u003C\u002Fem>-tube trapping the whole tail.\n  - p: |\n      The same ideas generalize to \u003Cstrong>metric spaces\u003C\u002Fstrong>, where\n      distance alone defines open sets, compactness, and completeness, and\n      the theorems of the real line reappear in their natural setting.\n  - p: |\n      The reward is judgment: you stop asking whether a calculation looks\n      right and start knowing precisely which hypotheses make it true.\n---\n\nReal analysis rebuilds calculus from the ground up, starting from what the real\nnumbers actually are. The sequence opens with the completeness of the reals — the\nleast-upper-bound property that separates the continuum from the rationals — then\ndevelops sequences, series, and the Cauchy criterion that characterizes\nconvergence intrinsically. From there it moves to the topology of the real line\nand of general metric spaces: open and closed sets, compactness, and\nconnectedness, the abstract home of the theorems that follow. Continuity,\ndifferentiation, and Riemann integration are each given their proper\nepsilon-delta footing, culminating in the mean-value and fundamental theorems.\nA careful treatment of pointwise versus uniform convergence explains exactly when\nlimits, derivatives, and integrals may be interchanged, and the notes close with\nthe calculus of several variables. Organized after Rosenlicht, with Lebl and\nShkoller as the working texts.\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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etic-trajectories":235,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":276,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":277,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":278,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":279,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":280,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":281,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":282,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":283,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":284,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":209,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":285,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":286,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":287,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":288,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":289,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":290,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":291,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":292,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":227,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":226,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":293,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":294,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":295,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":296,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":297,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":298,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":299,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":300,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":301,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":253,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":251,"\u002Felectricity-and-magnetism":302,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":303,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":304,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":305,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":306,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":307,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":308,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":309,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":310,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":311,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":161,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":312,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":313,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":165,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":314,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":315,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":316,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":317,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":318,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":319,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":320,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":321,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":322,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":323,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":324,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":325,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":326,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":327,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":328,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":329,"\u002Flinear-algebra\u002Forthogonality-least-squares\u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and the Real Number System",1,"foundations",[993,998,1003,1009],{"title":994,"path":995,"lessonNumber":990,"topics":996,"summary":997},"Sets, Logic, and Functions","\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions",[989],"The working language of analysis: quantifiers and the proof patterns (contrapositive, contradiction, induction), sets and their operations, relations and equivalence classes, and functions with their images, injections, surjections, and bijections. Cardinality is measured by bijection, and Cantor's theorem that no set surjects onto its power set forces uncountable sets to exist.\n",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"Ordered Fields and the Completeness Axiom","\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness",[989],"The real numbers are the unique ordered field with the least-upper-bound property. The field and order axioms, the exact failure of the rationals (no supremum for the set of rationals below √2), and completeness as the defining axiom of ℝ lead to the first consequences: the existence of √2, the Archimedean property, and the density of ℚ in ℝ.\n",{"title":1004,"path":1005,"lessonNumber":1006,"topics":1007,"summary":1008},"Absolute Value, Bounded Sets, and Inequalities","\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds",3,[989],"The absolute value turns the order on ℝ into a notion of distance, with the triangle inequality as the estimate underlying most later proofs. Covered: its algebra, the triangle and reverse-triangle inequalities, and the extension of the sup\u002Finf vocabulary from sets to bounded functions.\n",{"title":1010,"path":1011,"lessonNumber":1012,"topics":1013,"summary":1014},"Intervals, Uncountability, and Decimals","\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability",4,[989],"Intervals are classified, and ℝ is proved uncountable two ways: a nested-interval construction and the decimal diagonal argument. Decimal expansions are built as suprema of truncations, which pins the source of their non-uniqueness (the 0.4999… equals 0.5000… identity) and the identification of the rationals with the eventually-repeating expansions. The middle-thirds Cantor set is an uncountable set of measure zero.\n",{"module":1016,"moduleNumber":16,"slug":1017,"lessons":1018},"Sequences and Series","sequences-series",[1019,1024,1029,1034,1039,1045],{"title":1020,"path":1021,"lessonNumber":990,"topics":1022,"summary":1023},"Sequences and Their Limits","\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits",[1016],"A sequence is a function on the natural numbers; it converges to a limit when its terms eventually stay within any prescribed tolerance of that number. The epsilon-M definition fixes the order of the quantifiers, and from it the limit is unique, every convergent sequence is bounded, and only the tail matters. Divergence to plus or minus infinity records terms that outgrow every bound.\n",{"title":1025,"path":1026,"lessonNumber":16,"topics":1027,"summary":1028},"Limit Laws and Monotone Convergence","\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone",[1016],"Limits commute with sums, products, quotients, roots, and absolute values and preserve non-strict inequalities, so a limit can be assembled from the limits of its parts without returning to epsilon and M. The squeeze lemma transfers a limit through two envelopes; the monotone convergence theorem produces a limit from boundedness alone; and the ratio test settles the geometric and factorial standard limits.\n",{"title":1030,"path":1031,"lessonNumber":1006,"topics":1032,"summary":1033},"Subsequences, Limit Superior, and Bolzano–Weierstrass","\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass",[1016],"A bounded sequence need not converge, but it always has convergent subsequences, and its terms cluster between two extreme values. The limit superior and inferior are the limits of the tail suprema and infima; they always exist for a bounded sequence, coincide exactly when it converges, and are its largest and smallest subsequential limits. Bolzano–Weierstrass extracts a convergent subsequence from boundedness alone.\n",{"title":1035,"path":1036,"lessonNumber":1012,"topics":1037,"summary":1038},"Cauchy Sequences and the Completeness of the Reals","\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness",[1016],"The Cauchy criterion tests convergence without knowing the limit: a sequence converges exactly when its terms eventually all lie within any tolerance of one another. Cauchy sequences are bounded, in the reals Cauchy and convergent are equivalent, and this completeness property is interchangeable with the least-upper-bound axiom — the single feature that separates the real line from the rationals.\n",{"title":1040,"path":1041,"lessonNumber":1042,"topics":1043,"summary":1044},"Series and Convergence Tests","\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence",5,[1016],"A series converges when its sequence of partial sums does, so every fact about sequences transfers. Geometric and telescoping series sum in closed form; the n-th term test rejects series whose terms miss zero, though the harmonic series shows the converse fails; and the comparison test against the geometric and p-series benchmarks settles most nonnegative-term series.\n",{"title":1046,"path":1047,"lessonNumber":1048,"topics":1049,"summary":1050},"Absolute Convergence, the Ratio and Root Tests, and Rearrangements","\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement",6,[1016],"Absolute convergence is the strong form of convergence that permits free manipulation; conditional convergence is fragile. Absolute convergence implies convergence, and the ratio and root tests detect it by comparison with the geometric series. The alternating series test supplies conditionally convergent series, Riemann's theorem rearranges any of them to any sum, and Mertens' theorem multiplies series when at least one converges absolutely.\n",{"module":1052,"moduleNumber":1006,"slug":1053,"lessons":1054},"Metric Spaces and Topology","metric-spaces",[1055,1060,1065,1070,1075],{"title":1056,"path":1057,"lessonNumber":990,"topics":1058,"summary":1059},"Metric Spaces, Norms, and Examples","\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms",[1052],"A metric is a function $d(x,y)$ obeying four axioms: nonnegativity, identity of indiscernibles, symmetry, and the triangle inequality. The Euclidean, taxicab, sup, discrete, and great-circle metrics all qualify, as does the sup metric on $C[a,b]$. Every norm induces a metric, and strongly equivalent metrics share the same open sets.\n",{"title":1061,"path":1062,"lessonNumber":16,"topics":1063,"summary":1064},"Open and Closed Sets, Interior, Closure","\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets",[1052],"Open sets are those in which every point has room to move; closed sets are their complements. From the single ball construction come the topology axioms (arbitrary unions, finite intersections), the interior, closure, and boundary of a set, and the fact that openness is always relative to the ambient space.\n",{"title":1066,"path":1067,"lessonNumber":1006,"topics":1068,"summary":1069},"Convergence, Cauchy Sequences, and Completeness","\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness",[1052],"The $\\varepsilon$-$N$ definition of a limit transfers verbatim to any metric space once $|x-y|$ is replaced by $d(x,y)$. Convergent sequences characterize closed sets and closures; Cauchy sequences and completeness capture spaces with no missing limits, with $\\mathbb{R}^n$ and $C[a,b]$ complete and $\\mathbb{Q}$ and $(0,1]$ not.\n",{"title":1071,"path":1072,"lessonNumber":1012,"topics":1073,"summary":1074},"Compactness","\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness",[1052],"A set is compact if every open cover has a finite subcover. In a metric space this is equivalent to sequential compactness and to being complete and totally bounded. Compact sets are closed and bounded; the Heine–Borel theorem gives the converse in $\\mathbb{R}^n$ but nowhere else in general.\n",{"title":1076,"path":1077,"lessonNumber":1042,"topics":1078,"summary":1079},"Connectedness","\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness",[1052],"A space is connected when it cannot be split into two nonempty open pieces. The connected subsets of $\\mathbb{R}$ are precisely the intervals, path- connectedness gives a constructive sufficient condition, and connectedness is a topological invariant preserved by continuous maps, the fact behind the intermediate value theorem.\n",{"module":1081,"moduleNumber":1012,"slug":1082,"lessons":1083},"Limits and Continuity","continuity",[1084,1089,1094,1099,1104,1109],{"title":1085,"path":1086,"lessonNumber":990,"topics":1087,"summary":1088},"Limits of Functions","\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions",[1081],"The limit of a function at a point is an epsilon–delta condition pinning one value L as the target of f(x) as x approaches c, mirroring the sequence definition with distance replacing index. It is stated only at cluster points of the domain, is unique when it exists, and reduces to sequential limits through the Heine criterion. The algebra of limits and one-sided limits follow from that reduction.\n",{"title":1090,"path":1091,"lessonNumber":16,"topics":1092,"summary":1093},"Continuous Functions","\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions",[1081],"A function is continuous at c when its limit there equals its own value, lim f(x) = f(c). The epsilon–delta and sequential forms agree; sums, products, quotients, and compositions of continuous functions are continuous; and the failures split into jump, Dirichlet, popcorn, and removable types. The topological reading is that preimages of open sets are open.\n",{"title":1095,"path":1096,"lessonNumber":1006,"topics":1097,"summary":1098},"Extreme and Intermediate Value Theorems","\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt",[1081],"On a closed bounded interval a continuous function attains an absolute maximum and minimum (the extreme value theorem, compactness preserved by continuity) and takes every value between its endpoint values (the intermediate value theorem, connectedness preserved). Both proofs run through Bolzano–Weierstrass and bisection, and yield root-finding, existence of k-th roots, and fixed-point theorems.\n",{"title":1100,"path":1101,"lessonNumber":1012,"topics":1102,"summary":1103},"Uniform Continuity","\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity",[1081],"Uniform continuity strengthens continuity by demanding one delta that works at every point of the domain, not a delta re-chosen at each point. It separates x^2 on a compact interval from x^2 on the whole line and from 1\u002Fx near zero; continuity on a closed bounded interval is automatically uniform; uniformly continuous functions preserve Cauchy sequences and extend to endpoints; and Lipschitz continuity is the strongest of the three, through its secant-slope bound.\n",{"title":1105,"path":1106,"lessonNumber":1042,"topics":1107,"summary":1108},"Continuity on Metric Spaces","\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces",[1081],"The epsilon–delta definition used only distances, so continuity transfers to maps between metric spaces by replacing absolute values with the two metrics. In this generality continuity still admits a sequential form, preserves compactness and connectedness, is uniform on a compact domain, and reads topologically as preimages of open sets being open, the formulation that defines homeomorphisms.\n",{"title":1110,"path":1111,"lessonNumber":1048,"topics":1112,"summary":1113},"Limits at Infinity and Monotone Functions","\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone",[1081],"Treating infinity as a cluster point extends the epsilon–delta limit to x approaching plus or minus infinity, giving horizontal asymptotes and infinite limits. For monotone functions the one-sided limits always exist as suprema and infima, the discontinuities are jumps and at most countably many, the continuity is equivalent to the image being an interval, and a strictly monotone function always has a continuous inverse.\n",{"module":1115,"moduleNumber":1042,"slug":1116,"lessons":1117},"Differentiation","differentiation",[1118,1123,1128,1133],{"title":1119,"path":1120,"lessonNumber":990,"topics":1121,"summary":1122},"The Derivative","\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative",[1115],"The derivative is the limit of the difference quotient, the slope the secant lines approach as the second point slides into the first. Differentiability forces continuity; linearity and the product, quotient, and chain rules follow from the definition; and a continuous function can fail to be differentiable, as the absolute value does at the origin.\n",{"title":1124,"path":1125,"lessonNumber":16,"topics":1126,"summary":1127},"The Mean Value Theorem","\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem",[1115],"A relative extremum in the interior forces the derivative to vanish; Rolle's theorem and the mean value theorem turn that local fact into global control. The sign of the derivative fixes monotonicity, a bounded derivative yields a Lipschitz bound, and Darboux's theorem shows derivatives have the intermediate value property even where they are discontinuous.\n",{"title":1129,"path":1130,"lessonNumber":1006,"topics":1131,"summary":1132},"Taylor's Theorem","\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem",[1115],"Taylor's theorem generalizes the mean value theorem: an n-times differentiable function is matched near a point by a degree-n polynomial, with a Lagrange remainder that names the error exactly through one higher derivative. Iterating the mean value theorem proves it; the second-derivative test is the order-one case; and a smooth non-analytic bump separates a Taylor series from the function it fails to represent.\n",{"title":1134,"path":1135,"lessonNumber":1012,"topics":1136,"summary":1137},"The Inverse Function Theorem in One Variable","\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d",[1115],"A nonzero derivative certifies a local inverse and fixes its slope. A strictly monotone differentiable function has a differentiable inverse whose derivative is the reciprocal of the original; the inverse function theorem removes the monotonicity hypothesis, and the reciprocal formula constructs nth roots and the logarithm's derivative, failing exactly where the derivative vanishes.\n",{"module":1139,"moduleNumber":1048,"slug":1140,"lessons":1141},"The Riemann Integral","riemann-integration",[1142,1147,1152,1157,1162],{"title":1143,"path":1144,"lessonNumber":990,"topics":1145,"summary":1146},"Partitions, Darboux Sums, and Integrability","\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral",[1139],"The Riemann integral is defined by trapping the area under a bounded function between under- and over-estimates. Partitions cut the domain into strips; lower and upper Darboux sums bracket the area; refining a partition tightens the bracket. A function is integrable exactly when the bracket can be made arbitrarily thin, and the tagged Riemann-sum limit gives the same number.\n",{"title":1148,"path":1149,"lessonNumber":16,"topics":1150,"summary":1151},"Which Functions Are Integrable","\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes",[1139],"The Cauchy criterion certifies whole classes of functions as integrable. Continuous functions are integrable because uniform continuity makes every oscillation cap small; monotone functions are integrable because their caps telescope to a single total jump; bounded functions with finitely many discontinuities are integrable by isolating the bad points. The Dirichlet function fails, and the Lebesgue criterion names the exact boundary.\n",{"title":1153,"path":1154,"lessonNumber":1006,"topics":1155,"summary":1156},"Properties of the Integral","\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral",[1139],"The integral is a linear, order-preserving, additive operator on the integrable functions. It splits across subintervals, respects inequalities, bounds the size of a function by the integral of its absolute value, and preserves products. The mean value theorem for integrals identifies the integral with an attained average height on a fixed rectangle.\n",{"title":1158,"path":1159,"lessonNumber":1012,"topics":1160,"summary":1161},"The Fundamental Theorem of Calculus","\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem",[1139],"The fundamental theorem ties the integral to the derivative in two forms. The evaluation form computes a definite integral from any antiderivative; the differentiation form shows the area function has derivative equal to the integrand at points of continuity. Together they make differentiation and integration inverse operations, and yield integration by parts and change of variables.\n",{"title":1163,"path":1164,"lessonNumber":1042,"topics":1165,"summary":1166},"The Logarithm, Exponential, and Improper Integrals","\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper",[1139],"The integral defines transcendental functions. The logarithm is the area under 1\u002Ft, the exponential is its inverse, and their calculus properties follow from the fundamental theorem. Improper integrals extend integration to unbounded intervals and unbounded integrands as limits of proper integrals, with a p-test, a comparison test, absolute versus conditional convergence, and the integral test linking integrals to series.\n",{"module":1168,"moduleNumber":1169,"slug":1170,"lessons":1171},"Sequences and Series of Functions",7,"function-sequences",[1172,1177,1182,1187],{"title":1173,"path":1174,"lessonNumber":990,"topics":1175,"summary":1176},"Pointwise and Uniform Convergence","\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence",[1168],"A sequence of functions has two natural notions of limit. Pointwise convergence fixes each input and takes the limit of numbers; uniform convergence demands one rate that works for every input at once. The uniform norm turns the second into a statement about a single sequence of numbers, and the uniform Cauchy criterion and the Weierstrass M-test let us certify it.\n",{"title":1178,"path":1179,"lessonNumber":16,"topics":1180,"summary":1181},"Interchange of Limits: Continuity, Integration, Differentiation","\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits",[1168],"Passing to a limit inside a continuity statement, an integral, or a derivative is an interchange of two limits, and the two limits do not always commute. Uniform convergence licenses the first two swaps: the uniform limit of continuous functions is continuous, and the limit of the integrals is the integral of the limit. Differentiation needs uniform convergence of the derivatives, and counterexamples show why each hypothesis is required.\n",{"title":1183,"path":1184,"lessonNumber":1006,"topics":1185,"summary":1186},"Power Series and the Weierstrass Approximation Theorem","\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass",[1168],"A power series converges uniformly on every closed subinterval inside its radius of convergence, together with all of its derivatives. That makes it continuous, differentiable, and integrable term by term, so a power series defines an infinitely differentiable function. The Weierstrass approximation theorem then shows that polynomials come uniformly close to any continuous function on a closed bounded interval.\n",{"title":1188,"path":1189,"lessonNumber":1012,"topics":1190,"summary":1191},"Picard's Existence and Uniqueness Theorem","\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode",[1168],"The Banach fixed-point theorem says a contraction of a complete metric space has exactly one fixed point, found by iterating from any start. Applied to the space of continuous functions with the uniform norm, it proves Picard's theorem: a first-order differential equation with a Lipschitz right-hand side has a unique local solution. Picard iteration constructs that solution explicitly, and worked examples show the Lipschitz condition is not optional.\n",{"module":1193,"moduleNumber":1194,"slug":1195,"lessons":1196},"Functions of Several Variables (Introduction)",8,"several-variables",[1197,1202,1207,1212,1217],{"title":1198,"path":1199,"lessonNumber":990,"topics":1200,"summary":1201},"The Derivative of a Map ℝⁿ → ℝᵐ","\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn",[1193],"The derivative of a map between Euclidean spaces is the linear transformation of vanishing relative error, unique when it exists and represented in coordinates by the Jacobian matrix of partial derivatives. Differentiability forces continuity through a local Lipschitz bound. Existence of the partial derivatives alone does not suffice; continuity of the partials does.\n",{"title":1203,"path":1204,"lessonNumber":16,"topics":1205,"summary":1206},"Directional Derivatives, the Gradient, and the Chain Rule","\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule",[1193],"The directional derivative measures the rate of change of a scalar field along a chosen heading and equals the derivative applied to that direction. The gradient collects these into a vector that points along steepest ascent and sits orthogonal to level sets. The chain rule composes derivatives by multiplying Jacobians, and a mean value theorem holds for scalar fields but fails for vector-valued maps.\n",{"title":1208,"path":1209,"lessonNumber":1006,"topics":1210,"summary":1211},"Higher Derivatives, Taylor's Theorem, and Extrema","\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema",[1193],"Iterating the derivative gives a symmetric second derivative, the Hessian, whose mixed partials agree when they are continuous. Taylor's theorem expands a smooth map to any order with a Lagrange-type remainder, and at a critical point the definiteness of the Hessian decides between a local minimum, a local maximum, and a saddle.\n",{"title":1213,"path":1214,"lessonNumber":1012,"topics":1215,"summary":1216},"The Inverse and Implicit Function Theorems","\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems",[1193],"A nonlinear map with a nonsingular Jacobian is locally invertible, with the inverse's derivative given by the inverse matrix. The contraction mapping principle supplies the local inverse; the implicit function theorem then solves a system for some variables in terms of the rest whenever the relevant Jacobian block is invertible. Worked coordinate changes show both theorems in use.\n",{"title":1218,"path":1219,"lessonNumber":1042,"topics":1220,"summary":1221},"Multiple Integrals","\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals",[1193],"The Riemann integral of a bounded function over a closed rectangle in Euclidean space is built from Darboux upper and lower sums on a grid of subrectangles, with the same squeeze criterion that governs the one-variable integral. Continuous integrands are integrable, and a set of content zero can be ignored. Fubini's theorem evaluates a multiple integral as an iterated one in either order, and the indicator trick extends the theory to regions bounded by curves.\n",1786059478614]