Measurability of the max function.

Question

Could someone help me with the following problem?

Let $S$ be a compact metric space, $\mu$ be a Borel measure on a Souslin space, $f:S\times X\to\mathbb{R}$ be a function continuous in the first argument, and Borel measurable in the second argument. Show that there is a Borel measurable function $F:X\to S$ such that $F(x)$ is a point of maximum of $f(x,s)$ over $s\in S$.

I was given two hints:

1. We are looking for the function $F$ such $f(F(x),x)=\max\limits_{s\in S}f(x,s)$
2. We may use the fact that a Borel measurable mapping between Souslin spaces has a Borel measurable right inverse.

This question might seems as duplicate of the question Measurability of supremum over measurable set, but the answer looks like an overkill. I'm looking for a more down to Earth solution.