Real and Imaginary Part Projections are Continuous

Theorem
Define the real-valued functions $x, y: \C \to \R$ by:


 * $\forall z \in \C: x \left({z}\right) = \operatorname{Re} \left({z}\right)$


 * $\forall z \in \C: y \left({z}\right) = \operatorname{Im} \left({z}\right)$

Equip $\R$ with the usual Euclidean metric.

Equip $\C$ with the usual Euclidean metric.

Then both $x$ and $y$ are continuous functions.

Proof
Let $z \in \C$, and let $\epsilon \in \R_{ >0 }$.

Put $\delta = \epsilon$.

For all $w \in \C$ with $\left\vert{w - z}\right\vert < \delta$:

This equation shows that $\left\vert{x \left({w}\right) - x \left({z}\right)}\right\vert < \epsilon$, and $\left\vert{y \left({w}\right) - y \left({z}\right) }\right\vert < \epsilon$.

It follows by definition that $x$ and $y$ are both continuous.