Let $u: \mathbb{R}^2 \to \mathbb{R}$ be a differentiable function. Prove that if the complex function
$f(x + iy) = u(x,y) + iu(x,y)$
is analytic in $\mathbb{C}$ then it is a constant function.
Answer:
If $f$ is a analytic it satisfies the Cauchy Riemann equations. So $u_x = u_y$ and $u_x=-u_y$
This can only happen when $u_x$ and $u_y$ are equal $0$.
As the partial derivatives of $f$ are $0$, $f$ must be a constant function.
Is that correct?