Limit superior of a function at a point

Let \(A\subseteq\RR^n\), let \(f\colon A\to \RR\), and let \(y\in\RR^n\) be a limit point of \(A\). The limit superior of \(f\) at \(y\) is \[ \limsup_{x\to y} f(x) \coloneqq \lim_{\varepsilon\to 0^+} \sup f\qty{A \cap B_\varepsilon(y) \setminus\theset{y}} \in [-\infty,\infty], \] where \(B_\varepsilon(y)\) is the open ball of radius \(\varepsilon\) about \(y\); the limit exists because the supremum is nonincreasing as \(\varepsilon\) decreases.