Consider $f \in C^{1}(\mathbf{R})$.$ U_{1},U_{2}: [0,1] \rightarrow \mathbf{R}$ are $H^{1}(0,1)$ functions. My question is how to prove that:$\vert\vert f(U_{1})-f(U_{2}) \vert \vert_{H^{1}(0,1)} \leq C\vert\vert U_{1}-U_{2} \vert \vert_{H^{1}(0,1)}$, where C is a positive constant only depending on $\vert\vert U_{1}\vert \vert_{H^{1}(0,1)}$ and $ \vert\vert U_{2}\vert \vert_{H^{1}(0,1)} $?
My first guess is to use Taylor expansion, but I can only get $\vert\vert f(U_{1})-f(U_{2}) \vert \vert_{H^{1}(0,1)} \leq C(\vert\vert U_{1}-U_{2} \vert \vert_{H^{1}(0,1)}+1)$. I'm so thankful if somebody can help me out.