Continuity of a function of product spaces
Let $f: X \times Y \rightarrow Z $ such that.
$ \forall x_0 \in X , f_{x_0} : Y \rightarrow Z, y\mapsto f(x_0,y)$ is continuous.
$ \forall y_0 \in Y , f_{y_0} : X \rightarrow Z, x \mapsto f(x,y_0)$ is continuous.
How i can prove that $f$ is continuous?
Thanks in advance.
It isn't. Let $f:\Bbb R\times \Bbb R\to \Bbb R$ be the function
$(x,y)\mapsto\begin{cases}
0 &\text{ , if }x=y=0\\
\frac{xy}{x^2+y^2} &\text{ , else }
\end{cases}$
For any fixed $x_0$ the function $f_{x_0}$ is continuous. Since $f$ is symmetric in $x$ and $y$, $f_{y_0}$ is continuous for any $y_0$. But $f$ is not continuous in $x=y=0$. Consider the sequence $(x_n,y_n)=(1/n,1/n)$. It converges to $(0,0)$ along the diagonal and $f(x_n,y_n)=1/2$. But $f(\lim_{n\to\infty}(x_n,y_n))=f(0,0)=0$.