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:
“In todays world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.”
—Urie Bronfenbrenner (b. 1917)
“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 (19131960)
“No two people see the world exactly alike, and different temperaments will often apply the same principle, recognized by both, differently. Even one and the same person wont always maintain the same views and judgments: earlier convictions must give way to later ones.”
—Johann Wolfgang Von Goethe (17491832)