Question about continuity of a function to prove that it is Borel-measurable

Question

If $F(x,y)$ is a continuous function and I define $G(x)=\sup\{F(x,y) : y \in \mathbb{R} \}$. Will this function be continuous?

It would be really helpful because I'm trying to prove that $G$ is Borel- measurable.

Answer 1

Suppose $F(x,y)=0$ if $x<0$ or $y <0$, $\frac x y$ if $0\leq y<x$ and $1$ if $y \geq x$ then $F$ is continuous and $G(x)=0$ for $x <0$ and $G(x)=1$ for $x \geq 0$. Hence $G$ need not be continuous. However $G$ is measurable. We can write $\{x:G(x)<a\}=\cup_{y\in \mathbb q} \{x:F(x,y) <a\}$ which is a Borel set.