Definition:Continuity/Functional

Definition
Let $S$ be a set of mappings.

Let $y\in S$ be a mapping.

Let $J\sqbrk{y}:S\rightarrow \R$ be a functional.

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

Then $J\sqbrk{y}$ is said to be a continuous functional and is continuous at the point $y_{0}\in S$.