The supremum of a continuous mapping

Question

Let $E$ be a compact metric space and $f:E\times E\rightarrow (-\infty,0]$ a continuous mapping such that, $f(x,y)=0$ if and only if $x=y$ . Is it possible that the supremum is negative on the set: $E\times E-\Delta$, with $\Delta=\{(x,x),x\in E\}$?

Answer 1

Yes, it's possible: let $E$ be a finite discrete space and $$ f(x,y)=\begin{cases} -1 & x\ne y \\[4px] 0 & x=y \end{cases} $$ If $E$ is not discrete, then no: take a non isolated point $x$ and a sequence $(x_n)$ converging to $x$, with $x_n\ne x$ for every $n$.

Then $\sup\{f(x_n,x):n\in\mathbb{N}\}=0$.

Note that compactness plays no role.