 Hölder exponent greater than one forces constancy

If a real function on an interval obeys $|f(x)-f(y)|\le C|x-y{|}^{\alpha }$ 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 $|f(x)-f(y)|\le C|x-y{|}^{\alpha }{N}^{1-\alpha }\to 0$.