If f is differentiable at x = a, then f is also continuous at x = a. The converse is not true though: If a function is continuous at x = a, it is not necessarily differentiable at x = a.

Testing for differentiability

A function f with domain D is differentiable at x = a, if the following two conditions are met:

(1.) f is continuous at x = a

(2.) A left-hand derivative of f, and a right-hand derivative of f, both exist and are equal.

Worked Example

Prove that f(x) = |x| = x for and -x for x<0 is continuous but not differentiable at x = 0.

Exercise

Answers