Hölder exponent greater than one forces constancy
= Hölder exponent greater than one forces constancy
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{title2=$\alpha>1\Longrightarrow f\text{ constant}$}
If a real function on an interval obeys $|f(x)-f(y)|\leq C|x-y|^\alpha$ for finite $C$ and $\alpha>1$, its difference quotients tend to zero everywhere. The <mean value theorem> makes it constant. Equivalently, divide a segment into $N$ pieces to obtain $|f(x)-f(y)|\leq C|x-y|^\alpha N^{1-\alpha}\to0$.