Let $X, Y, Z$ be topological spaces and $f: X \times Y \rightarrow Z$ a function such that, for every fixed $x \in X$ and $y \in Y$, $f_x:Y \rightarrow Z$ and $f_y:X \rightarrow Z$ defined as $f_x(y)=f(x,y)$ and $f_y(x)=f(x,y)$
Show an example of a function $f$ such that $f_x$ and $f_y$ are continuous, but $f$ isn't.
I tried with some spaces like Sierpinski, some similar sets with 3 or 4 points, $\mathbb{R}$ and subsets of it, but couldn't find any that suffies this. Any ideas? Thanks
Take $X=Y=\mathbb R$, endowed with the usual topology, and take$$f(x,y)=\begin{cases}\frac{xy}{x^2+y^2}&\text{ if }(x,y)\neq(0,0)\\0&\text{ otherwise.}\end{cases}$$Each $f_x$ and each $f_y$ is continuous, but $f$ is discontinuous at $(0,0)$.