Continuity of Complex Functions

Continuity

Definition: Continuity of Complex Functions

Let $f:\mathcal{D}\subseteq \mathbb{C}\to \mathbb{C}$ be a complex function.

A complex function $f:\mathcal{D}\subseteq \mathbb{C}\to \mathbb{C}$ is continuous at $c\in \mathcal{D}$ if and only if its limit for $z\to c$ is equal to its value there.

$$\underset{z\to c}{\lim }f(z)=f(c)$$

We say that $f$ is continuous on $S\subseteq \mathcal{D}$ if it is continuous at each $z\in S$. Moreover, if $S=\mathcal{D}$, we just say that $f$ is continuous.

Theorem: Continuity of the Real and Imaginary Parts

Let $f:\mathcal{D}\subseteq \mathbb{C}\to \mathbb{C}$ be a complex function.

If $f$ is continuous at $c\in \mathbb{C}$, then its real part $\mathrm{Re}f$ and real part $\mathrm{Re}f$ part $\mathrm{Im}f$ are also continuous at $c$.

PROOF

TODO

Theorem: Continuity of Sum, Product and Division

Let $f$ and $g$ be complex function.

If $f$ and $g$ are continuous at $c\in \mathbb{C}$, then so are

• $\lambda f+\mu g$ for all $\lambda ,\mu \in \mathbb{C}$;

• $fg$;

• $f/g$, provided that $g(c)\ne 0$.

PROOF

TODO

Theorem: Continuity of Composition

Let $f$ and $g$ be complex function.

If $g$ is continuous at $c\in \mathbb{C}$ and $f$ is continuous at $g(c)$, then their composition $f\circ g$ is also continuous at $c$.

PROOF

TODO

Bibliography

1. N. H. Asmar, L. Grafakos, “Analytic Functions,” in Complex Analysis with Applications, Columbia, MO, USA: Springer, 2018