Question

If f is continuous at a, must f be differentiable at a?

1) Yes, f must be differentiable at a
2) No, f may or may not be differentiable at a
3) Cannot be determined
4) None of the above

Answer 1

No, f may or may not be differentiable at a.

Step-by-step explanation:

No, f may or may not be differentiable at a.

A function that is continuous at a point does not necessarily mean that it is differentiable at that point. Differentiability requires both continuity and the existence of a derivative. There are counterexamples where a function can be continuous but not differentiable at a certain point.

For example, consider the absolute value function, f(x) = |x|, which is continuous at x = 0 but not differentiable at x = 0. Therefore, option 2) No, f may or may not be differentiable at a, is the correct answer.