Limit superior of a function at an endpoint

For $f:(a,T)\to[-\infty,\infty]$ , with finite $a<T$ , the limit superior at $T$ from the left is

\limsup_{t\uparrow T}f(t) =\inf_{0<\delta<T-a}\ \sup_{T-\delta<t<T} f(t).

The suprema and infimum are interpreted in the extended real numbers. They may therefore equal either infinity.

Unbounded endpoint behavior

The lim sup is $+\infty$ exactly when $f$ is unbounded above in every left neighborhood of $T$ . Equivalently, one can choose $t_j\uparrow T$ with $f(t_j)\to+\infty$ . This does not require $f(t)\to+\infty$ along every approach.

Essential variant

For measurable functions, replacing each supremum by an essential supremum gives the essential lim sup. The ordinary version can depend on values at isolated points; the essential version ignores changes on null sets.