Determining a functions differentiability

Question

If a function has a point of discontinuity, such that the slope of tangents at points before and after that point are equal, will the function be differentiable?

Answer 1

Note that

$$\lim_{h\to 0} \frac {f(x+h)-f(x)}{h} =f'(x) $$

$$\implies \lim _{h\to 0} {f(x+h)-f(x)} =\lim _{h\to 0}h f'(x)=0$$

That is, if the function is differentiable at $x$ then it is also continuous at $x$.