Functions To Which Theorem Does Not Apply
The following examples show why the function domain must be closed and bounded in order for the theorem to apply. Each fails to attain a maximum on the given interval.
- ƒ(x) = x defined over [0, ∞) is not bounded from above.
- ƒ(x) = x / (1 + x) defined over [0, ∞) is bounded but does not attain its least upper bound 1.
- ƒ(x) = 1 / x defined over (0, 1] is not bounded from above.
- ƒ(x) = 1 – x defined over (0, 1] is bounded but never attains its least upper bound 1.
Defining ƒ(0) = 0 in the last two examples shows that both theorems require continuity on .
Read more about this topic: Extreme Value Theorem
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