Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point

Let \(D\) be a subset of real numbers \(\mathbb R\), \(a\in D\), and let \(f:D\to\mathbb R\) be continuous at $a$ with $f(a)\neq 0.$ Then $f$ is non-zero in a whole neighborhood of $a$, i.e. there exists a real number $\delta > 0$ such that $f(x)\neq 0$ for all $x\in D$ with $|x-a| < \delta.$

Proofs: 1

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