Weak-Strong, Lipschitz and uniform continuity

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 $$\mathbb{R}$$.

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.