Question

Given $f\left(z\right)=g\left(z\right)+h\left(z\right)$ where $f,g,h$ are complex valued function of a complex variable $z$. Which one of the following statements is TRUE?

• A. If $f\left(z\right)$ is differentiable at $z_o$, then $g\left(z\right)$ and $h\left(z\right)$ are also differentiable at $z_0$
• B. If $g\left(z\right)$ and $h\left(z\right)$ are differentiable at $z_0$, then $f\left(z\right)$ is also differentiable at $z_o$
• C. If $f\left(z\right)$ is continuous at $z_0$ then it is differentiable at $z_0$
• D. If $f\left(z\right)$ is diffeentiable at $z_0$, then so are its real and imaginary parts.

Answer 1

The correct option is B If $g\left(z\right)$ and $h\left(z\right)$ are differentiable at $z_0$, then $f\left(z\right)$ is also differentiable at $z_o$

Option $\left(A\right)$ is incorrect as $g$ and $h$ may indiviually be indifferentiable inspite of their sum being differentiable at a point.
Option $\left(B\right)$ is correct because if both are indivually differentiabe then their sum is definitely differentialble.
Option $\left(C\right)$ is incorrect as continuity doesn't ensure differentiability.
Option $\left(D\right)$ is incorrect as it is not correct to say that it $f$ is differentiable at $Z_o$ then real and imaginary parts of $f$ are differentiable at real and imaginary parts of $Z_o$