Definition:Minimum Value of Real Function/Local/Strict

Definition

Let $f$ be a real function defined on an open interval $\openint a b$.

Let $\xi \in \openint a b$.

$f$ has a strict local minimum at $\xi$ if and only if:

$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x > \map f \xi$

Also known as

A strict local minimum is also known as a strict relative minimum.

Some sources refer to this as a local minimum and do not consider the situation where $\forall x \in \openint c d: \map f x \le \map f \xi$.

Some sources assume local as given, and merely refer to this as a minimum or strict minimum.

Also see

• Definition:Local Minimum

• Definition:Strict Local Maximum
• Definition:Local Maximum

• Results about strict local minima can be found here.

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Maximum, Minimum and Point of Inflection
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): minimum (plural minima)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): minimum (plural minima)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): local minimum (minima)