Is derivative function a discontinued function as the limit says as h tends to 0 and nothing can be divided by 0?

Question

What does really the rate of change in function mean here; the real function must be like $\frac {F(x+h) - F(x) } h$ and not $\lim_{h\rightarrow 0} \frac {F(x+h) - F(x) } h$