Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point

(related to Definition: Continuous Functions at Single Real Numbers)

Let \(D\) be a subset of real numbers \(\mathbb R\), \(a\in D\), and let \(f,g:D\to\mathbb R\) be two function. Let $f$ be continuous at $a$ and let it be identical to $g$ in a neighborhood of $a$, i.e. for an (arbitrarily small) positive real number $\epsilon > 0$, let $f(x)=g(x)$ for all $x$ with $a a-\epsilon < x < a+\epsilon$. Then $g$ is continuous at $g$.

Proofs: 1

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