Recall that the essential range $ess.im \; f$ of a measurable function $f \in L^\infty(\mathbb{R})$ is a compact set. Denote by $f_k$ the restriction of $f$ to the interval $[-1/k,1/k]$, and $E_k$ the essential range of $f_k$, and let $E = \bigcap_{k=1}^{\infty} E_k$.

By Cantor's intersection theorem, $E$ is nonempty. This seems dangerously close to assigning function values to $f(0)$, which seems weird to me. Does this thing have a name and where can I find more about it? Or did I screw up somewhere?

Recall that the essential range $\operatorname{ess.im} f$ of a measurable function $f \in L^\infty(\mathbb{R})$ is a compact set. Denote by $f_k$ the restriction of $f$ to the interval $[-1/k,1/k]$, and $E_k$ the essential range of $f_k$, and let $E = \bigcap_{k=1}^{\infty} E_k$.

By Cantor's intersection theorem, $E$ is nonempty. This seems dangerously close to assigning function values to $f(0)$, which seems weird to me. Does this thing have a name and where can I find more about it? Or did I screw up somewhere?