Upper Bound of the $f(x,y)=\frac{|x^a-y^a|}{|x-y|}$, where $a$ is real, $0 < b\le x, y\le c$.

Question

$f(x,y)=\frac{|x^a-y^a|}{|x-y|}$, where $a$ is real, $0 < b\le x, y\le c$. I would like to find out this expression's upper bound (expressed in terms of a, b, c) so that I can kind of prove the continuity of a power function.