 Supremum can fail to be attained at an oscillatory endpoint

Define a bounded function on $[0,1]$ byIt is continuous away from zero, but its supremum is $1$ and it has no global maximum. For every $x>0$, $f(x)\le 1-x<1$, and $f(0)=0<1$. On the other hand,The strict pointwise bound proves nonattainment, while this sequence proves that no smaller upper bound works. The extreme value theorem fails to apply because of the discontinuity at zero. Defining an endpoint value below its oscillatory limit superior cannot repair that missing hypothesis.