1.10 Exploring Types of Discontinuities

Continuity requires three conditions

A function f is continuous at x = c if: (1) f(c) is defined, (2) the limit as x approaches c exists, and (3) that limit equals f(c). If any condition fails, the function has a discontinuity at that point.

Removable discontinuities

A removable discontinuity (hole) occurs when the limit exists at x = c but either f(c) is undefined or doesn't match the limit. This often happens with a common factor canceling in a rational function, like (x²-4)/(x-2) at x = 2.

Jump and infinite discontinuities

A jump discontinuity occurs when the left-hand and right-hand limits both exist but are unequal (common in piecewise functions). An infinite discontinuity occurs when the function approaches ±∞ near x = c, usually from a vertical asymptote where the denominator equals zero without canceling.

Key terms

Three conditions for continuity at x = c

f(c) is defined, the limit as x→c exists, and the limit equals f(c).

Removable discontinuity

A hole in the graph where the limit exists but doesn't match f(c), or f(c) is undefined; often from a canceling factor.

Jump discontinuity

Occurs when left-hand and right-hand limits both exist but are unequal, common in piecewise functions.

Infinite discontinuity

Occurs when f(x) approaches ±∞ near x = c, typically at a vertical asymptote.

How to fix a removable discontinuity

Redefine f(c) to equal the limit value at that point, making the function continuous there.

Try an AP-style question

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

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