(related to Proposition: Composition of Continuous Functions at a Single Point)
Let \((\xi_n)_{n\in\mathbb N}\), \(\xi\in D\) be a convergent real sequence with \(\lim \xi_n=a\) . By hypothesis, \(f:D\to\mathbb R\) is continuous at \(a\in D\). Therefore, we have \(\lim f(\xi_n)=f(a)\).
Also by hypothesis, \(g:E\to\mathbb R\) is continuous at \(f(a)\in E\). Therefore, we have \(\lim g(f(\xi_n))=g(f(a))\). It follows that the composition. \[g\circ f:D\to\mathbb R\] is continuous at \(a\).
