Let f : (0, 1) → R be a continuous function such that limx→0 f(x) = limx→1 f(x) = 0. Show that f achieves either an absolute minimum or an absolute maximum on (0, 1) (but perhaps not both).
Hint: A picture could help. One might also think "too bad the function isn’t defined and continuous in all of [0, 1], then we could apply the min-max theorem." However, isn’t that easily remedied?
Solution. I will give two solutions. The second solution, slightly modified, can used to prove that the same result holds if f : R → Ris continuous and limx→±[infinity] f(x) = 0.