$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$.

If $f(x_0) = g(x_0) $ for some point $x_0\in M$, is it true that $f$ is continuous at $x_0$? (I think it is true just using the definition of semi-continuous functions using $\limsup_{x\rightarrow y} f(x) \leq f(y)$ and $\liminf_{x\rightarrow y} g(x) \geq g(y)$)

What about the topology of the set of the continuous points of $f$? Or the topology of the set of $x$ such that $f(x) = g(x)$ (open or closed？)

Thank you!