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 $\left\vert{ w - z }\right\vert < \delta$, we have:

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 from definition of continuity that $x$ and $y$ are both continuous.