1.11 Defining Continuity at a Point

Three conditions for continuity

A function f is continuous at x = c if all three hold: (1) f(c) exists, (2) lim x→c f(x) exists (both one-sided limits are equal), and (3) lim x→c f(x) = f(c). If any condition fails, f is discontinuous at c.

Types of discontinuity

A removable discontinuity (hole) occurs when the limit exists but doesn't equal f(c), or f(c) is undefined. A jump discontinuity occurs when the left- and right-hand limits exist but differ. An infinite discontinuity occurs when f(x) approaches ±∞ near c, often from a vertical asymptote.

Checking continuity with one-sided limits

For piecewise functions, continuity at a boundary point c requires lim x→c⁻ f(x) = lim x→c⁺ f(x) = f(c). Compute each piece's limit separately, then compare to the function's defined value at c to confirm all three continuity conditions are satisfied.

Key terms

Definition of continuity at x = c

f(c) exists, lim x→c f(x) exists, and lim x→c f(x) = f(c).

Removable discontinuity

A hole in the graph where the limit exists but doesn't equal f(c), or f(c) is undefined.

Jump discontinuity

Occurs when the left- and right-hand limits both exist but are not equal to each other.

Infinite discontinuity

Occurs when f(x) approaches ±∞ near x = c, often due to a vertical asymptote.

Continuity check for piecewise functions

Verify lim x→c⁻ f(x) = lim x→c⁺ f(x) = f(c) at the boundary point.

Try an AP-style question

Which is NOT one of the three conditions for f to be continuous at x = a?

1. lim(x→a) f(x) = f(a)
2. f'(a) exists
3. lim(x→a) f(x) exists
4. f(a) is defined