Background
It is a standard and important fact in basic calculus/real analysis that a continuous function on a compact metric space is in fact uniformly continuous. That is, suppose $(X,d)$ is a compact metric space and $f\colon X \to\mathbb R$ is such that for every $x\in X$ and $\varepsilon>0$ there exists $\delta>0$ such that $d(x,y)<\delta$ implies $|f(x)-f(y)|<\varepsilon$. Then in fact, such a $\delta$ can be chosen independently of $x$.
Question
Does a similar statement hold regarding semi-continuous functions? For concreteness, let's consider upper semi-continuous functions, so suppose $(X,d)$ is compact and $f\colon X \to\mathbb R$ has the property that for every $x\in X$ and $\varepsilon >0$ there exists $\delta >0$ such that $d(x,y)<\delta$ implies $f(y) < f(x)+\varepsilon$. (Note the asymmetry of $x$ and $y$ in this definition.) Then is it true that $\delta=\delta(\varepsilon)$ can be chosen independently of $x$?
Reformulation
Given $\delta, \epsilon > 0$, consider the set$$X_\delta^\epsilon := \lbrace x\in X \mid f(y) < f(x) + \epsilon \text{ for every } y\in B(x,\delta) \rbrace.$$Then $f$ is upper semi-continuous if and only if $\displaystyle\bigcup_{\delta>0} X_\delta^\epsilon = X$ for every $\epsilon > 0$, and $f$ is uniformly upper semi-continuous if and only if this union stabilises -- that is, if for every $\epsilon > 0$ there exists $\delta>0$ such that $X_\delta^\epsilon = X$.