Definition:Bounded Mapping/Complex-Valued

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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