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:
“Mark the babe
Not long accustomed to this breathing world;
One that hath barely learned to shape a smile,
Though yet irrational of soul, to grasp
With tiny finger—to let fall a tear;
And, as the heavy cloud of sleep dissolves,
To stretch his limbs, bemocking, as might seem,
The outward functions of intelligent man.”
—William Wordsworth (1770–1850)
“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)
“He that will not apply new remedies, must expect new evils: for Time is the greatest innovator: and if Time, of course, alter things to the worse, and wisdom and counsel shall not alter them to the better, what shall be the end?”
—Francis Bacon (1561–1626)