TITLE: When does differentiability imply analyticity?
QUESTION [8 upvotes]: Suppose that a complex valued function $f:\mathbb{C}\to\mathbb{C}$ is complex differentiable at $z$, that is, $\lim\limits_{\Delta z \to 0}\frac{f(z+\Delta z)-f(z)}{\Delta z} = f'(z)$ exists and is finite. Suppose also that $f$ is continuous in some neighborhood of $z$ (or some open ball around $z$). Would this imply that $f$ is analytic at $z$ (or maybe in a neighborhood around it)? Can you give a counterexample? Can you think of a function that is analytic at a single isolated point?

REPLY [8 votes]: Let $f(z)=|z|^2=x^2+y^2$. $f$ is continuous on $\mathbb{C}$ (even $C^\infty$ as a function of the real variables $x$ and $y$), $f'(0)=0$ (in the complex sense), but $f$ is not complex differentiable at any $z\ne0$.