Consider a function $f:\mathbb{R}\to\mathbb{C},\ f=u+iv$ with $u,v\in\mathbb R$, which is is continuous with respect to the standard topology.
What is the weakest sufficient condition for the real and imaginary part of the function to be continuous as function $\mathbb{R}\to\mathbb{R}$? When does the converse fail?
Motivation: I expected the condition on $f$ to be continuous is the same as the conditions of $u$ and $v$, but I come to ask this because the following sections on the Wikipedia article on homolorphic functions: A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if f is continuous, u and v have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then f is holomorphic.
Why would they need to say $f$ is continuous if the real and imaginary part have derivatives, even?
Best Answer
It is true that $f$ is continuous if and only if the component functions are continuous (see Continuity of a vector function through continuity of its components for example).
Notice that your original function is $\mathbb R \to \mathbb C$ but in the wikipedia article they are talking about functions from $\mathbb C$ to itself. So these may be viewed as functions $\mathbb R^2 \to \mathbb R^2$. In dimensions two and higher, saying that the partial derivatives exist does not imply that the function is continuous (e.g. Pedro's answer).