If derivatives of a function exist, is the function continuous?

Question

I made this statement:

If the right and left derivatives of a function $f$ exist at each point of a set $A$ where it's defined, then the function $f$ is also continuous.

I made this assumption based on the fact that if $f$ is differentiable at a point, then $f$ is also continuous at a point. However, the statement doesn't say that is it continuous at a point. It just says that the function is continuous.

I have to write an essay regarding differentiation and continuity, so please tell me if what I wrote is right.

Answer 1

It is true that a function $f$ that is both left- and right-differentiable at a point $a$ is continuous at $a$.

However, the reasoning does not use the 'fact' that $f$ is differentiable at $a$, as that might be false. Just look at the function $f(x) = |x|$ at $x = 0$.

Instead, use the fact that left-(or right-)differentiability implies left-(or right-)continuity and that left- and right-continuity together imply continuity.