I have a question which might be fairly elementary, but I could not find an answer in the literature yet. Any pointers are very welcome :)
Let $X$, $Y$ and $Z$ be affine algebraic varieties. I have a map $f:X\times Y \rightarrow Z$ and I know that for fixed $x_0\in X$, the map $Y\rightarrow Z, y \mapsto f(x_0,y)$ is a morphism. Also, for fixed $y_0\in Y$, the map $X\rightarrow Z, x \mapsto f(x,y_0)$ is a morphism. I regard $X\times Y$ as the product variety of $X$ and $Y$.
Is $f$ itself a morphism then? Can I prove this by stating a few known results from algebraic geometry? (Pointers to literature would be very helpful). Or do I have to prove it "by hand"? Or is the statement even false?