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

  1. If $f(z)$ is differentiable at $z_{0}$, then $g(z)$ and $h(z)$ are also differentiable at $z_{0}$.
  2. If $g(z)$ and $h(z)$ are differentiable at $z_{0}$, then $f(z)$ is also differentiable at $z_{0}$.
  3. If $f(z)$ is continuous at $z_{0}$, then it is differentiable at $z_{0}$. 
  4. If $f(z)$ is differentiable at $z_{0}$, then so are its real and imaginary parts
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