Timewise-max function

This article defines a scalar function on a manifold, viz a function from the manifold to real numbers. The scalar function may be intrinsic or defined in terms of some other structure/functions

Definition

Let $u:\mathbb{R}\times M\to \mathbb{R}$ be any continuous function. Then the timewise-max function of $u$ is a map $f:\mathbb{R}\to \mathbb{R}$ defined by:

$f(t)={sup}_{x\in M}u(t,x)$

When $M$ is a compact manifold, we can replace $sup$ by $max$.

Related notions

• Timewise-min function
• Timewise-average function