Continuity of complex function

A complex function $f$ is continuous at $z_0$ if

${\lim }_{z\to z_0}f(z)=f(z_0).$

That is, the limit exists, $f$ is defined at $z_0$, and the two agree. $f$ is continuous on a set if it is continuous at every point of the set.

Reduction to real-variable continuity

Writing $f(z)=u(x,y)+iv(x,y)$, $f$ is continuous at $z_0=x_0+i{y}_0$ iff both $u$ and $v$ are continuous as functions of two real variables at $(x_0,y_0)$. So complex continuity reduces to a pair of standard multivariable continuity statements.

What’s continuous

• Constants and the identity $f(z)=z$.
• $\bar{z}$, $|z|$, $\mathrm{Re}(z)$, $\mathrm{Im}(z)$.
• Polynomials (sums of $a_n{z}^n$): continuous everywhere.
• Rational functions $p(z)/q(z)$: continuous everywhere $q(z)\ne 0$.
• $e^z$, $\sin z$, $\cos z$: continuous everywhere.
• Sums, products, compositions, and quotients (with nonzero denominator) of continuous functions.

A continuous-but-not-differentiable function

$f(z)=\bar{z}$ is continuous everywhere on $\mathbb{C}$ — both $u(x,y)=x$ and $v(x,y)=-y$ are continuous as functions of $(x,y)$.

Yet $\bar{z}$ is complex differentiable nowhere. Continuity is much weaker than differentiability in the complex setting.

This is in striking contrast to the real-variable case, where continuous-but-nowhere-differentiable functions (like the Weierstrass function) are pathological and rare. In the complex setting, continuous-but-nowhere-differentiable is typical — any smooth function involving $\bar{z}$, $\mathrm{Re}(z)$, $\mathrm{Im}(z)$, or $|z|$ in an essential way will be of this kind.

Differentiability implies continuity

The converse — continuity does not imply differentiability — fails dramatically. But complex differentiability does imply continuity:

${\lim }_{z\to z_0}(f(z)-f(z_0))={\lim }_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}(z-z_0)=f^{\prime }(z_0)\cdot 0=0.$

So any analytic function is continuous on its domain of analyticity.

In context

Continuity is the prerequisite for asking deeper questions. The Complex derivative requires more than continuity. Cauchy’s theorem and the Cauchy integral formula require analyticity. But you can’t even begin to define integration of a complex function along a contour without continuity along that contour — so this is the entry-level regularity.