How should this limit superior from a proof of the Lebesgue Differentiation Theorem be interpreted? On page 104 of Stein and Shakarchi's Real Analysis, in their proof of the Lebesgue Differentiation Theorem, the following limit superior appears: $$ \limsup_{m(B) \to 0,\,x \in B} \bigg\vert\frac{1}{m(B)}\int_B f(y)\,dy - f(x)\bigg\vert, $$ where $f : \mathbb{R}^d \to \mathbb{R}$ is an integrable function and $B$ is an open ball in $\mathbb{R}^d$. I understand what the limit superior of a sequence is and what the limit superior of a function is. I'm guessing that this limit superior is the limit superior of a function, but I'm not sure. How should this limit superior be interpreted?

Fix $x$. For a given $\epsilon>0$, let $$ g(x,\epsilon):=\sup\left\\{\bigg\vert\frac{1}{m(B)}\int_B f(y)\,dy - f(x)\bigg\vert: x\in B\hbox{ and } m(B)<\epsilon\right\\} $$ As $\epsilon$ decreases to $0$, $g(x,\epsilon)$ also decreases, to some limit. This limit is the $\limsup$ you seek. (I presume that $B$ is to be a ball.)