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

 $\ds \cmod {\map \Re w - \map \Re z}$ $=$ $\ds \cmod {\map \Re w +i\map\Im w- \map \Re z-i\map\Im z+i\map\Im z-i\map\Im w}$ $\ds$ $\le$ $\ds \cmod {\map \Re w +i\map\Im w- \map \Re z - i \map \Im z}+\cmod{i\map\Im z-i\map\Im w}$ Triangle Inequality for Complex Numbers $\ds$ $\le$ $\ds \cmod {\map \Re w +i\map\Im w- \map \Re z - i \map \Im z}$ modulus is always non-negative $\ds$ $=$ $\ds \cmod{w-z}$ $\ds$ $<$ $\ds \delta$ $\ds$ $=$ $\ds \epsilon$

and

 $\ds \cmod {\map \Im w - \map \Im z}$ $=$ $\ds \cmod i \cmod {\map \Im w - \map \Im z}$ $\cmod i=1$ $\ds$ $=$ $\ds \cmod {i\map \Im w - i\map \Im z}$ Complex Modulus of Product of Complex Numbers $\ds$ $=$ $\ds \cmod {\map \Re w +i\map\Im w- \map \Re z-i\map\Im z+\map\Re z-\map\Re w}$ $\ds$ $\le$ $\ds \cmod {\map \Re w +i\map\Im w- \map \Re z - i \map \Im z}+\cmod{\map\Re z-\map\Re w}$ Triangle Inequality for Complex Numbers $\ds$ $\le$ $\ds \cmod {\map \Re w +i\map\Im w- \map \Re z - i \map \Im z}$ modulus is always non-negative $\ds$ $=$ $\ds \cmod{w-z}$ $\ds$ $<$ $\ds \delta$ $\ds$ $=$ $\ds \epsilon$

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

$\blacksquare$