Solutions depending on something continuously Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational 
 inequality is the following problem: find a $u \in K$ satisfying 
$$
\langle A(u),v-u \rangle \geq \langle F,v-u \rangle \text{  }\text{ for all } v \in K
$$
What would it mean to say ‘the solution of $u$ depends on $F$ continuously‘?Also how is this usually shown? Thanks.


What would it mean to say ‘the solution of $u$ depends on $F$ continuously‘

It means that for every $F$ and every $\epsilon>0$ there exists $\delta>0$ such that for every functional $G$ with $\|F-G\|<\delta$ we have $|\tilde u-u\|<\epsilon$, where $\tilde u$ is the solution of the variational inequality with $G$ instead of $F$.

how is this usually shown? 

By proving that every $\tilde u$ with $\|\tilde u-u\|\ge \epsilon$ fails the inequality for some $v\in K$.
