2.1Limits and Continuity

Updated 24 Jun 2026Chapter (PDF)

[2.1.1]Definition(Limit)#

A function $f: A \to B$ is said to have a limit at some $a \in \Lim(A)$ (denoted $\lim_{x \to a} f(x) = A$ ) if, for every $\epsilon > 0$ , there is some $\delta > 0$ such that

|x - a| < \delta \implies |f(x) - f(a)| < \epsilon.

Immediately from the definition, we see that for a function whose domain is an open subset of the complex numbers, it follows that

\lim_{x \to a} f(x) = A \implies \lim_{x \to a} \bar{f(x)} = \bar{A}.

The usual notion of continuity is lifted from real analysis.

[2.1.2]Definition(Derivative)#

A function $f: A \to B$ is said to be differentiable at some $a \in \Lim(A)$ if

\lim_{x \to a} \frac{f(x) - f(a)}{x - a}

exists. Then $f'(a)$ takes the value of this limit.

[2.1.3]Example(Derivatives of Functions)#

There are four classes of functions to consider in the context of complex analysis.

Real-to-Real. For a function $f: \bR \to \bR$ , the theory of differentiation is lifted from real analysis.
Real-to-Complex. Let $f: \bR \to \bC$ . Indeed, we see that $f(t) = x(t) + iy(t)$ , and from multivariate analysis, we recall $f'(t) = x'(t) + iy'(t)$ . There is nothing new here.
Complex-to-Real. Let $f: \bC \to \bR$ . From multivariate analysis, for a limit to exist, a limit through all paths must exist and be equivalent. If $f$ is differentiable at some $a \in \bC$ , note that approaching the difference quotient from the real axis produces a purely real number, whereas approaching the difference quotient from the imaginary axis produces a purely imaginary number. Thus $f'(a) = 0$ , and so $f' \equiv 0$ at points it is defined on.
Complex-to-Complex. This is what we devote our time in complex differentiation towards. It is far less trivial than the other cases.