Continuity of Exponential Function

Proof (related to "Continuity of Exponential Function")

Let \(a\in\mathbb R\). We have to show that the exponential function \(\exp:\mathbb R\to \mathbb R\) is continuous, formally
\[\lim_{x\to a}\exp(x)=\exp(a).\]
Let \((x_n)_{n\in\mathbb N}\) be any convergent real series with \(\lim_{n\to\infty} x_n=a\). We have then \(\lim(x_n-a)=0\). Together with the result \(\exp(0)=1\) it follows
\[\lim_{n\to \infty}\exp(x_n-a)=1.\]
Because of the non-zero property of the exponential function \(\exp(x)\neq 0\) for all \(x\in\mathbb R\), and because of the functional equation of the exponential function we can conclude that
\[1=\lim_{n\to \infty}\exp(x_n-a)=\frac{\lim_{n\to \infty}\exp(x_n)}{\lim_{n\to \infty}\exp(a)}=\lim_{n\to \infty}\frac{\exp(x_n)}{\exp(a)}.\]
\[\exp(a)=\lim_{n\to \infty}\exp(x_n).\]
In the last step we have used the formula for the quotient of convergent real sequences.

References

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983

Discussion