Real and Imaginary Part Projections are Continuous

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


 * $\forall z \in \C: \map x z = \map \Re z$


 * $\forall z \in \C: \map y z = \map \Im z$

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 $\cmod {w - z} < \delta$:

This equation shows that $\cmod {\map x w - \map x z} < \epsilon$, and $\cmod {\map y w - \map y z} < \epsilon$.

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