Definition:Continuity/Functional

Definition
The functional $J[y]$ is said to be continuous at a point $y_{0} \in S$ if

$ \forall \epsilon\in\R_{>0}:\exists\delta\in\R_{>0}:\left\vert{y-y_0}\right\vert<\delta\implies\left\vert{J[y]-J[y_0]}\right\vert<\epsilon$