True or False

If f is continuous at x = c, then f is differentiable at x = c.

False

False.

To determine whether this statement is true or false, we need to understand the definitions of continuity and differentiability.

A function f is continuous at x = c if the following three conditions are satisfied:
1. f(c) is defined (the value of the function at x = c exists).
2. The limit of f(x) as x approaches c exists.
3. The limit of f(x) as x approaches c is equal to f(c).

On the other hand, a function f is differentiable at x = c if the following condition is satisfied:
1. The derivative of f(x) at x = c exists.

While it is true that if a function is differentiable at x = c, it must also be continuous at x = c, the converse is not necessarily true.

Counterexample:
Consider the absolute value function f(x) = |x|. This function is continuous at x = 0 since f(0) is defined, and the limit of f(x) as x approaches 0 exists and is equal to f(0). However, the function is not differentiable at x = 0 because the derivative does not exist at that point.

Therefore, the statement "If f is continuous at x = c, then f is differentiable at x = c" is false. Being continuous at a point does not guarantee differentiability at that point.