Definition:Maximum Value of Real Function/Absolute

Definition

Let $f: \R \to \R$ be a real function.

Let $f$ be bounded above by a supremum $B$.

It may or may not be the case that $\exists x \in \R: \map f x = B$.

If such a value exists, it is called the (absolute) maximum of $f$ on $S$, and that this (absolute) maximum is attained at $x$.

Also known as

An absolute maximum is also known as a maximum value, or just a maximum if there is no need to distinguish it from a local maximum.

Also see

• Definition:Local Maximum

• Definition:Absolute Minimum
• Definition:Local Minimum

• Results about absolute maxima can be found here.

Linguistic Note

The plural form of maximum is maxima.

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 7.13$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): absolute maximum or minimum
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): maximum (plural maxima)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): absolute maximum or minimum
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): maximum (plural maxima)