Does partial differentiation preserve continuity in another variable?

Question

Let $f = f(x, y): D \rightarrow \mathbb{R}$, $D \subseteq \mathbb{R}^2$, be a continuous function such that $\partial f/\partial x$ exists. For a fixed $x$, does it follow that $y \mapsto (\partial f/\partial x)(x, y)$ is still continuous? Can you present a proof or give a counterexample?

Answer 1

Consider $f(x,y)=\begin{cases}x^2y\sin\frac1{xy}&\text{if }x\ne 0\land y\ne 0\\ 0&\text{if }x=0\lor y=0\end{cases}$. It's easily checked that $f$ is continuous, that $$\frac{\partial f}{\partial x}(x,y)=\begin{cases}2xy\sin\frac1{xy}-\cos\frac1{xy}&\text{if }x\ne 0\land y\ne 0\\ 0&\text{if }x=0\lor y=0\end{cases}$$

and that $\frac{\partial f}{\partial x}(c,\bullet)$ is discontinuous in $0$ for all $c\ne 0$.