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:
“The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
—Cyril Connolly (1903–1974)
“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)
“Children and savages use only nouns or names of things, which they convert into verbs, and apply to analogous mental acts.”
—Ralph Waldo Emerson (1803–1882)