= Interior-value criterion for an attained maximum
Let $f$ be continuous on $(a,b]$, bounded above, and suppose its <limit superior> at $a$ from the right is at most $L$. If some $c\in(a,b]$ has $f(c)>\max\{L,f(a)\}$, then $f$ attains its maximum on $[a,b]$. A sufficiently short initial interval has all values below $f(c)$. On the remaining closed interval, the <extreme value theorem> supplies a maximum, which is consequently global. This can prove attainment despite an oscillatory discontinuity at one endpoint.
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