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
Famous quotes containing the words functions, theorem and/or apply:
“Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
—Friedrich Nietzsche (1844–1900)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“We all want to be happy, and we’re all going to die.... You might say those are the only two unchallengeably true facts that apply to every human being on this planet.”
—William Boyd (b. 1952)