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Proposition: Composition of Continuous Functions at a Single Point
Let \(f:D\to\mathbb R\) and \(g:E\to\mathbb R\) be real functions with \(f(D)\subseteq E\) such that \(f\) is continuous at \(a\in D\) and \(g\) is continuous at \(f(a)\in E\). Then their composition.
\[g\circ f:D\to\mathbb R\]
is continuous at \(a\).
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- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983