\begin{definition}[Definition:Argument of Complex Number]
Let $z = x + i y$ be a complex number.
An '''argument of $z$''', or $\arg z$, is formally defined as a solution to the pair of equations:
:$(1): \quad \dfrac x {\cmod z} = \map \cos {\arg z}$
:$(2): \quad \dfrac y {\cmod z} = \map \sin {\arg z}$
where $\cmod z$ is the modulus of $z$.
From Sine and Cosine are Periodic on Reals, it follows that if $\theta$ is an '''argument''' of $z$, then so is $\theta + 2 k \pi$ where $k \in \Z$ is ''any'' integer.
Thus, the '''argument''' of a complex number $z$ is a continuous multifunction.
{{explain|what does it mean for a multifunction to be continuous?}}
\end{definition}