Continuity

$f(z)$ is continuous at $z_0$ iff:

$$\lim_{z\to z_0} f(z) = f(z_0)$$ $$\iff \forall{\epsilon>0}\; \exists{\delta>0}\; \forall{x}\; \big(\,|z-z_0|<\delta\implies{\lvert f(z)-f(z_0)\rvert<\epsilon}\,\big)$$

Conditions

For $f$ to be continuous on $z_0$, all these conditions are required.

• $f$ is defined at $z_0$
• $\lim_{z\to z_0} f(z)$ exists
• $\lim_{z\to z_0} f(z) = f(z_0)$

Properties

If $f,g$ are continuous at $z_0$, these functions are also continuous at $z_0$:

• $\text{Re}(f)$
• $\text{Im}(f)$
• $\lvert f \rvert$
• $f\pm g$
• $fg$
• $\frac{f}{g}$ where $g\neq 0$
• $f(g(z))$