AP Calculus BC Practice Test

Which of the following is NOT a condition for f to be continuous at c?

f differentiable at c f(c) is defined lim_{x->c} f(x) exists lim_{x->c} f(x) = f(c)

Correct Answer:

f differentiable at c

Explanation:

Continuity at a point means the function value at that point exists and nearby values of the function approach the same number. Concretely, you need three things: f(c) is defined, the limit as x approaches c of f(x) exists, and that limit equals f(c). Differentiability at c is not required for continuity; in fact, differentiable functions are continuous, but a function can be continuous without being differentiable at that point. For example, f(x) = |x| is continuous at 0, so the three continuity conditions hold, yet it is not differentiable at 0. Thus, differentiability at c is not a condition for continuity.

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