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

• A. f differentiable at c
• B. f(c) is defined
• C. lim_{x->c} f(x) exists
• D. lim_{x->c} f(x) = f(c)

Answer: (A)

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.