# A question on Hölder condition.

Suppose $$f:U\subset\mathbb{R}^n\to \mathbb{R}^{m}$$ satisfies the Hölder condition, i.e. there are constant $$\alpha>0$$ and $$L>0$$ such that $$\|f(x) - f(y)\|\leq L\cdot \|x - y\|^{\alpha},\quad \mbox{ for all } x,y\in \mathbb{R}^{n}.$$ It is well known that if $$\alpha>1$$ then $$f$$ is differentiable and constant.

There is an example of a continuous Hölder application $$f:U\subset\mathbb{R}^{n}\to \mathbb{R}^{m}$$ that satisfies the conditions below?

• $$n>1$$, $$\alpha>1$$ and $$f$$ is not constant;
• $$f$$ is not differentiable at all points of the sequence $$\{x_{k}\}_{k=1}^{\infty}\subset U$$;
• $$\{x_{k}\}_{k=1}^{\infty}\subset U$$ has an infinite number of accumulation points;
• $$f$$ is not differentiable at all accumulation points of the sequence;
• Did you mean to write $\alpha > 0$ on the first line and $\alpha > 1$ on the third line? Jul 11 at 20:07
• @JoseAvilez , I fixed the mistake. Thanks. Jul 11 at 20:16

In any dimension, as long as $$U$$ is an open set, then an $$\alpha$$-Hölder continuous function with $$\alpha>1$$ will be (locally) constant: The condition $$\dfrac{\| f(x)-f(y)\|}{\| x-y\|} \leq L\| x-y\|^{\alpha-1},\qquad x\neq y,$$ implies that the derivative of $$f$$ at any given $$x\in U$$ is zero (since the zero linear transformation works in having a limit for the quotient on the left).