A question about maximum theorem

Question

These days, I am learning maximum theorem. There are two statements about this theorem.

Wikipedia: link

Berge's book: Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity

If $\phi$ is a continuous numerical function in $Y$ and $\Gamma$ is a continuous mapping of $X$ into $Y$ such that, for each $x$, $\Gamma x \neq \varnothing$, then the numerical function $M$ defined by $M(x)=\max \{\phi(y) \mid y \in \Gamma x\}$ is continuous in $X$ and the mapping $\Phi$ defined by $\Phi x=\{y \mid y \in \Gamma x, \phi(y)=M(x)\}$ is a u.s.c. mapping of $X$ into $Y$.

My questions:

1. Is $f$ (Wikipedia) (or $\phi$ (Berge)) a univariate or multivariate function?

2. What is the difference between a compact-valued correspondence (C, Wikipedia) and a continuous mapping ($\Gamma$, Berge)?

Thanks a lot for any help.