Show, using the mean value theorem and given $|f(x)-f(y)| ≤ |x-y|^3$ and $x, y ∈ [0,1]$, that $f(x)$ has to be constant.

Question

Show, using the mean value theorem and given $|f(x)-f(y)| ≤ |x-y|^3$ and $x, y ∈ [0,1]$, that $f(x)$ has to be constant.

For which $a > 0$ does the same inference apply, given $|f(x)-f(y)| ≤ |x-y|^a$?