Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain

Let \(D\subseteq \mathbb R\) and let \(f,g:D\to\mathbb R\) be two real functions, which are continuous on their whole domain \(D\) and let \(\lambda\in\mathbb R\). Then the functions

\[\begin{array}{rcl} f + g:D&\to&\mathbb R\\ \lambda f:D&\to&\mathbb R\\ f\cdot g:D&\to&\mathbb R\\ \end{array}\]

are continuous on \(D\). Moreover, if \(g(a)\neq 0\) for all \(a\in D\), then the function

\[\begin{array}{rcl} \frac fg:D&\to&\mathbb R \end{array}\]

is also continuous on \(D\).

Proofs: 1

Proofs: 1 2

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983