Continuity

The Direct Substitution Property computes a limit of a polynomial or rational function by plugging in. The functions for which substitution works are the continuous ones: those whose graph has no holes, jumps, or breaks, so the curve can be drawn without lifting the pen.

Continuity at a point

The equation encodes three separate requirements; continuity fails if any one of them fails:

• $f(a)$ is defined — that is, $a$ lies in the domain of $f$.
• $\underset{x\to a}{\lim }f(x)$ exists — the two-sided limit is a finite number.
• The two agree — the limit equals the value.

If $f$ is defined near $a$ but is not continuous there, $f$ is discontinuous at $a$. Continuity captures the idea that a small change in $x$ produces only a small change in $f(x)$: the point $(x,f(x))$ slides toward $(a,f(a))$ with no jump.

The three ways continuity fails

Discontinuities come in three geometric types, distinguished by which requirement above breaks.

The types and their signatures:

• Removable. The limit exists, but $f(a)$ is undefined or unequal to it, leaving a single-point hole. Redefining $f$ at that one point repairs continuity. The function $f(x)=(x^2-x-2)/(x-2)$ has a removable discontinuity at $2$, where it is undefined though ${\lim }_{x\to 2}f(x)=3$.
• Jump. The one-sided limits both exist but differ, so the two-sided limit does not. The Heaviside function and every step function jump at their transitions.
• Infinite. A one-sided limit is $+\infty$ or $-\infty$, so the graph runs off to a vertical asymptote. The function $1/x^2$ has one at $0$.

Deciding the type reduces to checking the three requirements in turn: whether $f(a)$ is defined, whether the limit exists, and whether they match.

One-sided and interval continuity

At an endpoint of a domain, only one side is available, so continuity is defined by a one-sided limit.

For example, $f(x)=1-\sqrt{1-x^2}$ is continuous on $[-1,1]$: at interior points the limit equals the value, and at the endpoints it is continuous from the right at $-1$ and from the left at $1$, which is all the domain permits.

Building continuous functions

Verifying continuity from the definition every time would be tedious. A few theorems let complicated continuous functions be recognized at a glance.

This follows from the corresponding Limit Laws. Combined with the base cases ${\lim }_{x\to a}c=c$ and ${\lim }_{x\to a}x=a$, it certifies whole families at once.

A continuous outer function commutes with a limit: it can be moved inside.

Composing two continuous functions keeps continuity, which makes nested expressions tractable.

With these theorems, a function such as $F(x)=\sin (x^2+1)$ is continuous everywhere because it is a composite of the continuous $\sin$ with the continuous polynomial $x^2+1$; no limit calculation is required. Likewise $G(x)=(x^2+7)/(x-1)$ is continuous everywhere except $x=1$, the lone zero of its denominator.

The domain of continuity is read off directly from the pieces and where the denominator vanishes.

A piecewise function joined at a boundary can be forced continuous by tuning a parameter so the two pieces meet.

The Intermediate Value Theorem

Continuity on a closed interval has a consequence that is obvious to the eye and indispensable in practice: a continuous curve cannot get from one height to another without passing through every height in between.

Continuity is essential; a function with a jump can step over $N$ without ever attaining it. The theorem is an existence result: it guarantees a solution without producing one.

Locating roots

Setting $N=0$ turns the theorem into a root-existence test: a continuous function that changes sign across an interval must have a root inside. Repeating the test on the half that still straddles zero — bisection — traps the root in ever-shorter intervals.

The same sign-change test works on equations that no algebra solves, once they are rearranged so one side is zero.

The sign-change-and-bisect idea is the seed of the numerical root-finding refined in Newton's Method.