Let $\;f:\mathbb R \rightarrow \mathbb R^m\;$ and $\;g:\mathbb R^m \rightarrow \mathbb R\;$. Assume $\;g\;$ is uniform continuous in $\;\{ \vert f \vert \le M\;\}\;$, i.e.$\;\forall \varepsilon \gt 0\;$ $\exists \;δ\gt 0\;$ :
$\;\vert g(f_1)-g(f_2) \vert \le \varepsilon\;$, $\; \vert f_1-f_2 \vert \le δ\;$ for $f_1,f_2 \in \{ \vert f \vert \le M\;\}\;(1)$ ($\;M\;$ some positive constant).
If in addition, each $\;f\in \{ \vert f \vert \le M\;\}\;$ satisfies a Holder condition:
$\; \vert f(x_1)-f(x_2) \vert \le L {\vert x_1 -x_2 \vert}^{\frac {1}{2}}\;(2)$ for some $\;L\gt 0\;$
then I wonder if it is possible somehow to claim $\;\vert g(f(x_1))-g(f(x_2)) \vert \le \varepsilon\;$ for $\; L {\vert x_1 -x_2 \vert}^{\frac {1}{2}}\le δ\;$.....?
EDIT: Consider the norm mentioned as the Euclidean norm in $\;\mathbb R^m\;$
To be more specific, I'm interested in finding the relation between $\;(1)\;$ and $\;(2)\;$ in order to prove the above claim. I have been trying a lot before I ask here but the fact that in condition $\;(1)\;$ I have the same variable with different functions while in condition $\;(2)\;$ I have the same function with different variables , confuses me a lot.
I would appreciate any help. Thanks in advance!