Definition:Continuity/Functional

Definition

Let $S$ be a set of mappings.

Let $y \in S$ be a mapping.

Let $J \sqbrk y: S \to \R$ be a functional.

Suppose:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \size {y - y_0} < \delta \implies \size {J \sqbrk y - J \sqbrk {y_0} } < \epsilon$

Then $J \sqbrk y$ is said to be a continuous functional and is continuous at the point $y_0 \in S$.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 1.2$: Function spaces