Suppose $f(z)=u(x,y)+iv(x,y)$, where $z=x+iy$, for $x,y \in \mathbb{R}$. If u and v are real functions which are differentiable at all $(x,y) \in \mathbb{R}^2$. Then either show that it follows that $f(z)$ is differentiable as a function $f: \mathbb{C} \rightarrow \mathbb{C}$ or find a counterexample.
My first thought is that $f$ is differentiable and to use the Cauchy Reimann equations. However, I'm a little stuck and would appreciate any help. My logic is that if u and v are real and differentiable then they are continuous and the partial derivatives exist. It follows that the partial derivatives of $f$ exist. However, how do we know if the partial derivatives are continuous let alone whether or not the Cauchy Reimann equations hold?