# Urgent Help!

The idea for the first parts is that if $\epsilon > 0$ and $x$ is a fixed point then we can take $\delta > 0$ small enough such that $|x-y|^{\alpha} < \epsilon$ for all $|x-y| < \delta$. Therefore, $|f(x)-f(y)| \leq |x-y|^{\alpha} < \epsilon$. Thus, $f$ is continous.
In the second part if $\alpha > 1$ then $|f(x)-f(y)| \leq |x-y|^{\alpha}$ then $| [f(x)-f(y)]/[x-y] | \leq |x-y|^{\alpha - 1}$. Notice the LHS has form of a derivative. Thus, what can you conclude?