In our lecture of stationary differential equations with the theory of monotone operators we introduced the concept of weak-strong continuity with sequences:
Defintion (weak-strong continuity) Let $V$ be a reflexive Banach space. A (not not necessarily linear) operator $A: V \to V^*$ ($V^*$ is the continuous dual space) is called weak-strong continuous if $$ v_n \rightharpoonup v \text{ in } V \implies A v_n \to Av \text{ in } V^* $$ for all sequences $(v_n)_{n \in \mathbb{N}} \subset V$.
This is obviously a stronger requirement than continuity.
Since uniform and Lipschitz continuity are also stronger than continuity I wondered if there is a connection between them (as in: one implies the other under certain conditions) as i.e. $$\textrm{Lipschitz } \implies \textrm{ uniform } \implies \textrm{ continuous}$$on compact intervals in $\mathbb{R}$.
I think there is no connection between this properties in general.
First, let $A=Id$. This is a nice Lipschitz continuous and uniformly continuous but not weak-strong continuous.
Second, for $V=\mathbb R$, $A(x)=x^2$ is weak-strong continuous but neither Lipschitz nor uniformly continuous.