Proof
(related to Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments)
 Let $D\subset\mathbb R$ be a subset.
 By hypothesis, $f:D\to\mathbb R$ is a continuous function at an $a\in D.$
 By definition, this means that for every convergent real sequence $(x_n)_{n\in\mathbb N}$ with $x_n\in D$, $a\in D$ and $\lim_{n\to\infty}x_n=a$ we have that $\lim_{n\to\infty}f(x_n)=f(a).$
 On the other hand, we have that $f(\lim_{n\to\infty} x_n)=f(a),$ following the hypothesis, that we have a concrete real sequence $(x_n)_{n\in\mathbb N}$ given with $\lim_{n\to\infty} x_n=a.$
 Thus, we have that $\lim_{n\to\infty}f(x_n)=f(\lim_{n\to\infty} x_n).$
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983