Derivatives

If a function is continuous at x=c, is it always differentiable at x=c?

No

Consider the following function:

f(x)=-x for all x<0

f(x)=x for all x≥0

At x=0, the limit from the left (0) equals the limit from the right (0), and the function is continuous.

But the function it is not differentiable at x=0.

The graph has a corner at x=0, so a tangent line cannot be drawn there.

Study Sheet - Derivatives