Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments

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

Let $D\subset\mathbb R$ be a subset and let $(x_n)_{n\in\mathbb N}$ be a convergent real sequence with $x_n\in D$, $a\in D$ and $\lim_{n\to\infty}x_n=a.$ If a function $f:D\to\mathbb R$ is continuous at $a\in D,$ then $$\lim_{n\to\infty} f(x_n)=f(\lim_{n\to\infty} x_n)=f(a).$$

Roughly speaking, it is possible to "exchange the limits" between the limit of function values and the respective function value of the limit of function arguments.

Proofs: 1

Proofs: 1

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