Can you extend differentiability in $\mathbb{R}^2$ of real and imaginary parts of a function to say the function is entire in $\mathbb{C}$

Question

Given a function $u(x,y)$ that is harmonic on $\mathbb{R}^2$ we know that there must exist a unique (up to constant additive) harmonic conjugate of u(x,y), called $v(x,y)$ that is also harmonic on $\mathbb{R}^2$. Therefore there will exist a function $f(z)=u(x,y)+i v(x,y)$ that is differentiable on $\mathbb{R}^2$.

From here I want to claim that $f(z)$ is also entire on $\mathbb{C}$. This makes sense to me intuitively but I am having a bit of trouble formalizing the concept.

Answer 1

Yes, it is. Because $f$ is a holomorphic from $\mathbb C$ into $\mathbb C$, and therefore analyic. So, it's an entire function.