Continuity

_x → a f(x) = f(a) — no holes, no jumps, no blow-ups.

The idea

Continuity is the hypothesis nearly every theorem of calculus begins from. Informally, a function is continuous when we can draw its graph without lifting the pen.

To make that precise at a single point $a$, separate two numbers. One is the value $f(a)$, whatever the function is defined to be there. The other is the number $f(x)$ approaches as $x$ closes in on $a$ — the limit $\lim_{x \to a} f(x)$. Nothing forces these to agree: a function can approach one height and be defined at another, approach a height without being defined at $a$ at all, or approach different heights from the two sides and so have no limit.

Continuity is the condition that none of that happens.

Definition (Continuity at a point).

A function $f$ is continuous at $a$ when $\lim_{x \to a} f(x) = f(a).$

The equation demands three things — the limit exists, $f(a)$ is defined, and the two agree. Each can fail on its own, and each failure has a name: the limit exists but the value differs (a removable discontinuity), the two one-sided limits disagree (a jump), or the values grow without bound (an infinite discontinuity).

Ways to work on it

• Walkthrough. The three-part definition, and the taxonomy of discontinuities.
• Practice. Classify a function: continuous, removable, jump, or infinite.
• Hardest. Choose the constant that makes a piecewise function continuous.

Not sure where to start? Take the ten-question placement test.