# Real and Imaginary Part Projections are Continuous

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

 $\ds \cmod {\map \Re w - \map \Re z} + \cmod {\map \Im w - \map \Im z}$ $=$ $\ds \cmod {\map \Re w - \map \Re z} + \cmod {i \paren {\map \Im w - \map \Im z} }$ as $\cmod i = 1$ $\ds$ $\le$ $\ds \cmod {\map \Re w - \map \Re z + i \map \Im w - i \map \Re z}$ Triangle Inequality for Complex Numbers $\ds$ $=$ $\ds \cmod {w - z}$ $\ds$ $<$ $\ds \epsilon$

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.

$\blacksquare$