1.10 Types of Discontinuities

Three types

A removable discontinuity is a hole: the limit exists but doesn't equal f(c) (or f(c) is undefined). A jump discontinuity has unequal left and right limits (common in piecewise functions). An infinite discontinuity happens at a vertical asymptote.

Fixable or not

Only a removable discontinuity can be 'fixed' by redefining one point. Jumps and infinite discontinuities cannot be patched.

Key terms

Removable discontinuity

A hole: the limit exists but ≠ f(c) (or f(c) undefined).

Jump discontinuity

Left and right limits differ (a step).

Infinite discontinuity

Occurs at a vertical asymptote; values blow up.

Hole vs. asymptote

A hole is removable; an asymptote is infinite.

Fixable

Only removable discontinuities can be patched by redefining a point.

Try an AP-style question

What type of discontinuity does f(x) = (x² − 1)/(x − 1) have at x = 1?

1. Infinite
2. None — f is continuous there
3. Jump
4. Removable