Suppose we have two sets $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a continuous function $f:X\times Y\to \mathbb{R}$ such that, for each fixed $x\in X$, the function $f_x=f(x,\cdot)$ is in $C_0(Y)$, the space of continuous functions vanishing at infinity. My question is the following: given a compact set $K\subset X$, is the restriction of the function to the set $K\times Y$ uniformly continuous?
My intuition tells me that this is true, although I haven't been able to show it. I tried to get to a contradiction, but to finish my argument, I need some kind of uniform continuity.
Any hints or suggestions will be appreciated. Thanks in advance