Differentiability and Continuity
If a function is differentiable at point x = a, then the function is continuous at x = a.
The reciprocal may not be true, that is to say, there are functions that are continuous at a point which, however, may not be differentiable.
Examples
Study the continuity and differentiability of the following functions:
First, study the continuity at x = 0.
The function is not continuous, therefore it is not differentiable.
First, study the continuity at x = 0.
This function is continuous, so the differentiability can be studied.
It is not differentiable at x = 0.
f(x) = x2 at x = 0.
The function is continuous at x = 0, so the differentiability can be studied.
At x = 0, the function is continuous and differentiable.
