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 | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \cmod {\map \Re w + i \map \Im w - \map \Re z - i \map \Im z}\) | Complex Modulus is Non-Negative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cmod {w - z}\) | ||||||||||||
| \(\ds \) | \(<\) | \(\ds \delta\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \epsilon\) |
and
| \(\ds \cmod {\map \Im w - \map \Im z}\) | \(=\) | \(\ds \cmod i \cmod {\map \Im w - \map \Im z}\) | as $\cmod i = 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cmod {i \map \Im w - i \map \Im z}\) | 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 | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \cmod {\map \Re w + i \map \Im w - \map \Re z - i \map \Im z}\) | Complex Modulus is Non-Negative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cmod {w - z}\) | ||||||||||||
| \(\ds \) | \(<\) | \(\ds \delta\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \epsilon\) |
It follows by definition that $x$ and $y$ are both continuous.
$\blacksquare$
Notes
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This theorem can be considered a special case of Continuous Mapping to Product Space.
Suppose we let $z = \map \Re z + i \map \Im z$ be a complex number.
We can now identify the complex number $z$ with the ordered pair $\tuple {\map \Re z, \map \Im z} \in \R^2$, where $R^2$ is the Cartesian product $\R \times \R$.
The functions $x$ and $y$ can now be considered as projections on the co-ordinates.