Talk about the relation between differentiability and continuity on a real function and its derivative.

In a real 2-Dimensional function f(x) on the X-Y plane, we have the following relations between these concepts: i) f'(x) is continuous if and only f(x) is differentiable; in fact, the continuity of f'(x) ensures that there are no points where the derivative tends to infinity, or has a possible multiple value. (picture as additional explanation) ii) f(x) differentiable does not imply f(x) continuous, since we may have a function that is shifted up at a certain point, so it keeps to be differentiable, since there is no double derivative at that point, but the limits of x that tends to that point are different. (picture that function using a grapher) iii) f(x) continuous does not imply f(x) differentiable. In fact a simple counter example could be f(x)=|x|. At x=0, f(x) is continuous, checkable using the definition. But the derivative assumes a double value at x=0, f'(0)=1 and f'(0)=-1. Therefore we found a counter-example.