Concept preview

What you will be able to do

1. 01Define the three types of discontinuity — removable, jump, and infinite — in terms of one-sided limits and the function value.
2. 02Classify a discontinuity from a graph, a formula, or a piecewise definition by comparing $\lim_{x\to c^-}f$, $\lim_{x\to c^+}f$, and $f(c)$.
3. 03Identify the $x$-value and type of each break, and explain which discontinuities can be repaired and which cannot.

Three common ways a graph can fail continuity at $x=c$ are removable, jump and infinite discontinuities; oscillation is another possibility, and each has a distinct signature. A removable discontinuity is a single missing or misplaced point — a hole in an otherwise smooth curve. A jump discontinuity is a step, where the curve arrives at one height from the left and departs from a different height on the right. An infinite discontinuity is a vertical asymptote, where the curve shoots off to $+\infty$ or $-\infty$ near the point. Naming which one you are looking at is the goal of this lesson, and the tool for doing it is always the same trio: $\lim_{x\to c^-}f(x)$, $\lim_{x\to c^+}f(x)$, and $f(c)$.