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Lemma: Functions Continuous at a Point and NonZero at this Point are NonZero 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 nonzero 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 $xa < \delta.$
Table of Contents
Proofs: 1
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983