# how to prove this inclusion abot the discontinuity set of a composed function?

Let $D_{\phi}$ be the set o discontinuity points of $\phi$. How can i prove that if $g\circ f$ makes sense then $D_{g\circ f} \subset D_{f} \bigcup f^{-1}(D_{g})$?

for $x \in D_{g\circ f} \Rightarrow x \in D_{f}$ is obvious but what about $f^{-1}(D_{g})$? the exercise do not give any information about $f$ but he use $f^{-1}$, can i assume that $f$ is a bijection?

The reason for having $x \in f^{-1}(D_g)$ is because $g \circ f$ is defined where $f$ is defined on $D_g$.