(related to Proposition: Compositions of Continuous Functions on a Whole Domain)
This is a simple corollary to the composition of continuous functions at a single point \(a\), because \(f:D\to\mathbb R\) is a real function continuous at every point \(a\in D\) and the real function \(g:f(D)\to\mathbb R\) is continuous in every point \(f(a)\in E\), by hypothesis. Thus, it is continuous on $D$.