Continuous Functions

The high-school slogan is that a function is continuous if its graph can be drawn without lifting the pen. That picture is useful and, near a jump, misleading. The rigorous definition took Bolzano, Cauchy, and Weierstrass to settle, and it is the limit definition with $L$ forced to equal $f(c)$: a function is continuous at $c$ when nudging the input near $c$ keeps the output near $f(c)$, with no exceptional value at $c$ to discard.

Two differences from the limit definition: the target is prescribed ($f(c)$, not some free $L$), and the window now includes $c$ (no $\setminus \{c\}$), which is harmless because $|f(c)-f(c)|=0<ϵ$ always holds. As with the limit, $\delta$ depends on both $ϵ$ and the point $c$; one need not find a single $\delta$ good for all $c$. Uniform continuity is the stronger demand of one $\delta$ good for every $c$ at once.

Three equivalent characterizations

An isolated point carries no constraint: a $\delta$ small enough to trap only $x=c$ makes the condition vacuous, so every $f$ is continuous there; in particular, every $f:\mathbb{Z}\to \mathbb{R}$ is continuous. The sequential characterization imports the sequence limit laws wholesale and is the form used most often below.

The same fact has a direct $ϵ$–$\delta$ proof that exposes the explicit $\delta (ϵ,c)$, showing how the window must shrink near the singularity.

Building continuous functions

Continuity is stable under every algebraic operation, so large classes of functions are continuous for one-line reasons.

Two consequences cover most functions met in practice:

• Polynomials are continuous on $\mathbb{R}$. Since $x\mapsto x$ and constants are continuous, repeated sums and products give continuity of $a_d{x}^d+\cdots +a_{1}x+a_0$.
• Rational functions are continuous wherever the denominator is nonzero, by the quotient rule.

Composition

Failure modes of continuity

When $f$ is not continuous at $c$, we call $c$ a discontinuity. The sequential characterization gives the most direct test: if some $x_n\to c$ has $\{f(x_n)\}$ not converging to $f(c)$, then $f$ is discontinuous at $c$.⁵

A single factor of $x$ localizes the misbehavior to the origin: multiplying the Dirichlet function by $x$ leaves it continuous at exactly one point.

A function can therefore be continuous on a dense set and discontinuous on another dense set at once.

Removable versus not. Let $g(x)=0$ for $x\ne 0$ and $g(0)=1$. Here ${\lim }_{x\to 0}g(x)=0$ exists but misses $g(0)$; redefining $g(0)=0$ repairs continuity, so this is a removable discontinuity. The jump function above is not removable: no value at $0$ can equal two different one-sided limits.

The topological reading

Continuity has a formulation with no $ϵ$, no $\delta$, and no sequences, phrased entirely in open sets.

This formulation carries over unchanged to metric spaces, where balls replace intervals but preimage of open is open stays verbatim. On a closed bounded interval, continuity forces two theorems that calculus uses without proof: a continuous function there attains its extremes and takes every intermediate value, the extreme and intermediate value theorems.