# Definition:Supremum Norm/Continuous on Closed Interval Real-Valued Function

## Definition

Let $I = \closedint a b$ be a closed real interval.

Let $\map \CC I$ be the space of real-valued functions continuous on $I$.

Let $f \in \map \CC I$.

Let $\size {\, \cdot \,}$ denote the absolute value.

Suppose $\sup$ denotes the supremum of real-valued functions.

Then the supremum norm over $\map \CC I$ is defined as

$\ds \norm {f}_\infty := \sup_{x \mathop \in I} \size {\map f x}$