$A,B$ are blocks in $ \mathbb{R}^n$, and $f: A \rightarrow B$ a continuous map, such that $ ||f(x)-f(y)|| > c||x-y||$ for all $x,y \in A $ and some $c>0$. If $g: B \rightarrow \mathbb{R}$ is integrable. Prove that $g\circ f:A \rightarrow \mathbb{R}$ is integrable.
My approach is since $f$ is continuous then the set $D_f$ where $f$ is not continuous is empty so $D_f$ has null measure and $D_g$ is integrable then $D_g$ has also null measure. As $D_{g\circ f}\subset D_g\cup D_f$ then $D_{g\circ f}$ has null measure. By Lebesgue's Theorem $g\circ f$ is integrable.