Does a continuous function always have a maximum or minimum?

The Extreme value theorem states that if a function is continuous on a closed interval [a,b], then the function must have a maximum and a minimum on the interval.

Why a continuous function on an open interval may not have a maximum or minimum?

Some functions fail to have a maximum or minimum inside a closed interval because they are not continuous, and a continuous function may not have a maximum or minimum if its domain is not confined within a closed interval.

Does the extreme value theorem guarantee the existence of an absolute maximum and minimum?

The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions. It states the following: If a function f(x) is continuous on a closed interval [ a, b], then f(x) has both a maximum and minimum value on [ a, b].

Do all continuous functions have a maximum?

Every continuous function f:[a,b]→R has a maximum. True; if f is continuous on a closed, bounded interval, then it will have a min and a max.

Do functions need to be continuous to have absolute max?

The extreme value theorem states that continuity on a closed interval is sufficient to ensure that the function attains a maximum and minimum. However, this condition is not necessary.

Do open intervals have a maximum?

h(x) = x, 0 < x ≤ 1. The function h is continuous and defined on an open interval. It has neither an absolute maximum value nor an absolute minimum value.

Does a function have to be continuous to have absolute max?

The Extreme Value Theorem says that if f(x) is continuous on the interval [a,b] then there are two numbers, a≤c and d≤b, so that f(c) is an absolute maximum for the function and f(d) is an absolute minimum for the function.

Where must a continuous function have an absolute maximum value on a closed interval a/b ]?

Theorem 1 If f is continuous on a closed interval [a, b], then f has both an absolute maximum value and an absolute minimum value on the interval. This theorem says that a continuous function that is defined on a closed interval must have both an absolute maximum value and an absolute minimum value.

Does a continuous function have a minimum?

The example above shows that a continuous function on an non-closed interval may not have an absolute max or min. When the interval is closed, if the function is not continuous, it may still not have have both an absolute max or min. has an absolute max but no absolute min.