Let $f\in C^1([0,1]\times\mathbb{R},\mathbb{R})$, and $g_n \in C^1([0,1],\mathbb{R})$ that converges uniformly towards $g \in C^1([0,1],\mathbb{R})$ : i.e $$||g_n-g ||_{\infty}\rightarrow0. \,\, (n\rightarrow +\infty).$$ Can we deduce that :for all $t\in [0,1]$ : $|f(g_n(t),t)-f(g(t),t)|\rightarrow0. \,\, (n\rightarrow +\infty)$ ?
My idea:
Since $f\in C^1$, then $f$ is locally lipschitz : there is $c>0$ such that : $$|f(g_n(t),t)-f(g(t),t)|\leq c ||g_n-g||_{\infty}.$$