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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;
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    $\begingroup$ Did you mean to write $\alpha > 0$ on the first line and $\alpha > 1$ on the third line? $\endgroup$ Jul 11 at 20:07
  • $\begingroup$ @JoseAvilez , I fixed the mistake. Thanks. $\endgroup$ Jul 11 at 20:16
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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).

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