The differentiability of a function

Let $w=f(z)$ be a complex function. Suppose $\Delta z \to 0$ implies $\Delta w \to 0$. If a derivative of an inverse function $z=f^{-1}(w)$ exists, does it follow that $w=f(z)$ has a derivative? (In my book, the author showed that a complex function $w=\sqrt z$ has a derivative by saying

1)$w=\sqrt z$ is continuous. (I omit the proof here)

2)The inverse function $z=w^2$ has a derivative.

3)$\displaystyle \frac{dw}{dz}=\frac{1}{\frac{dz}{dw}}=\frac{1}{2w}=\frac{1}{2\sqrt z}$

I want to know the theoretical background. Thanks in advance.