Continuity, Discontinuities and Piecewise Functions

The three-part definition

$f$ is continuous at $x=c$ when all three hold:

1. $f(c)$ is defined;
2. $\lim_{x\to c}f(x)$ exists (both one-sided limits exist and are equal);
3. $\lim_{x\to c}f(x)=f(c)$.

On free-response, writing all three lines is worth points on its own — a bare "it's continuous" earns nothing.

Naming the failure

• Removable — the limit exists but disagrees with (or replaces a missing) $f(c)$. A hole.
• Jump — one-sided limits exist but differ. Typical of piecewise definitions.
• Infinite — at least one one-sided limit is $\pm\infty$. A vertical asymptote.

For a piecewise function with a parameter, continuity at the seam becomes an equation: set the left-hand expression evaluated at the seam equal to the right-hand expression evaluated at the seam, then solve.