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