Simple question about fourier series

Question

The problem is :

Let $f : R \rightarrow C$ a continuous function of period one. Supoose that f is Holder continuous with Holder exponent $\alpha > 1$. Then $f$ is constant.

I am trying to use the partials sums of the fourier series in a point $x$ and in a point $y$. But i am getting anywhere. Someone can give me a hint ?

thanks in advance

Answer 1

Hint: $$ \left|\frac{f(x+h)-f(x)}{h}\right|\le C|h|^{\alpha-1} $$ Note that there is an extension of Hölder continuity for $k\lt\alpha\lt k+1$ for $k\in\mathbb{N}$: $$ f\in C^k\quad\text{and}\quad\left|\Delta_h^kf(x)\right|\le C|h|^\alpha $$ which is equivalent to $f^{(k)}$ having Hölder exponent $\alpha-k$.