(related to Proposition: Continuity of Complex Exponential Function)
We have to show that the complex exponential function \(\exp:\mathbb C\to \mathbb C\) is continuous on whole \(\mathbb C\), formally for any \(a\in\mathbb C\), we have \[\lim_{x\to a}\exp(x)=\exp(a).\] Let \((x_n)_{n\in\mathbb N}\) be a convergent complex 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 complex exponential function \(\exp(x)\neq 0\) for all \(x\in\mathbb C\), and because of the functional equation of the complex 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 complex sequences.
