We know that we can study a compact operator between normed spaces $T: E \to F$ in two ways:
When $T$ is linear, it is only natural to ask that $\overline{T(A)}$ be compact in $F$ for all $A$ bounded in $E$;
When $T$ is nonlinear, it is natural to ask that $\overline{T(A)}$ be compact on $F$ for all $A$ bounded on $E$ and also that $T$ be a continuous map.
My question is: what is the motivation to ask for continuity for the non-linear case? guarantee the same classical characterizations for the linear case?