Is it true that there is a function that is differentiable at x= -2, but not continuous at x = -2?

Answer 1

All differentiable functions are continuous. However, not all continuous functions are differentiable.

Answer 2

No, differentiability implies continuity.

I am not sure why this answer was censored since it is precisely correct.

Any function which has a derivative at a point is continuous at that point.

You can easily prove this with the Mean Value Theorem.