# Definition:Bounded Mapping/Complex-Valued

## Definition

Let $f: S \to \C$ be a complex-valued function.

Then $f$ is **bounded** if and only if the real-valued function $\cmod f: S \to \R$ is bounded, where $\cmod f$ is the modulus of $f$.

That is, $f$ is **bounded** if there is a constant $K \ge 0$ such that $\cmod {f \paren z} \le K$ for all $z \in S$.

### Unbounded

Let $f: S \to \C$ be a complex-valued function.

Then $f$ is **unbounded** if and only if $f$ is not bounded.

That is, $f$ is **unbounded** if there does not exist a constant $K \ge 0$ such that $\cmod {f \paren z} \le K$ for all $z \in S$.

## Also see

- Complex Plane is Metric Space: this definition coincides with the definition of a bounded mapping to a metric space, using the standard metric on $\C$.