Analysis of differentiability after addition of complex valued functions It's given that $f(z)=g(z)+h(z)$, where $f,g,h$ are complex valued functions of a complex variable z. 
My questions are:


*

*If $g,h$ are differentiable at $z_{0}$, does that necessarily make $f$ differentiable at $z_{0}$ ?

*Is the converse of my previous question true  ?

If $g$ and $h$ are differentiable at $z_0$, $f \equiv g + h$ is differentiable at $z_0$ since
$$
\lim_{z\rightarrow z_{0}}\frac{f(z)-f(z_{0})}{z-z_{0}}=\lim_{z\rightarrow z_{0}}\frac{g(z)-g(z_{0})}{z-z_{0}}+\lim_{z\rightarrow z_{0}}\frac{h(z)-h(z_{0})}{z-z_{0}}=g^{\prime}(z)+h^{\prime}(z).
$$
The converse is not true.
Consider a function $g$ which is nondifferentiable at $z_0$, and let $h=-g$. Then, $f\equiv g+h=0$, which is clearly differentiable everywhere.
Note also that this does not have much to do with complex numbers.
