Infinitly differentiable hölder continuous function

Question

I want to find an example of an infinitly differentiable hölder continuous function with parameter $\alpha=1$, i.e. $f\in C^{0,1}$ for $f:A\to B\subset \mathbb{R}$, where $A$ is closed set in $\mathbb{R}^n$. For simplicity let $A=[a,b]^n$. Now if $n=1$, I would take the simple polynomial $f(x)=x$, which is infinetly differentiable and, since it is lipschitz, it is hölder continuous with parameter $\alpha =1$. My problem is that the functions range should be a subset of $\mathbb{R}$. So I do not see how to generalize this in higher dimension. For example the norm function, would be Lipschitz and it also has the right range, but is however not infinitely differentiable.

Thank you for your help.

hulik

Answer 1

Take $f$ a smooth bounded function with bounded derivative, and take $g(x):=f(\lVert x\rVert^2)$, where $\lVert \cdot\rVert$ is the Euclidian norm. Then $$|g(x)-g(y)|=\left|\int_0^1\partial_t g(tx+(1-t)y)dt\right|=\left|\sum_{j=1}^n2(x_i-y_i)f'(\lVert tx+(1-t)y\rVert^2)dt\right|\leq 2n\lVert x-y\rVert \sup_{v\in\Bbb R^n}|f'(v)|.$$ Example: $f(t)=e^{-t^2}$.