A function is continuous at $a$ when you can draw its graph through $x = a$ without lifting the pencil. The precise version is three conditions, checked in order: (1) $f(a)$ is defined; (2) $\lim_{x \to a} f(x)$ exists, meaning both one-sided limits exist and agree; (3) $\lim_{x \to a} f(x) = f(a)$, read aloud: the limit of f of x as x approaches a equals f of a. All three, or it is not continuous. The trap this lesson names is taking condition (1) for the whole thing: a function can have a perfectly good value at $a$ and still jump there, or still have its dot floating above a hole. Defined is one condition out of three.

Where continuity is free: polynomials are continuous everywhere; so are $\sin$, $\cos$, $e^x$ and $|x|$. A rational function $\dfrac{N(x)}{D(x)}$ is continuous wherever $D(x) \ne 0$, so its only candidates for trouble are the zeros of the denominator. A piecewise function is continuous inside each piece and needs checking only at the boundaries, where the three conditions earn their keep.