Identity Mapping is Continuous/Metric Space

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Theorem

Let $M = \struct {A, d}$ be a metric space.


The identity mapping $I_A: A \to A$ defined as:

$\forall x \in A: \map {I_A} x = x$

is a continuous mapping.


Proof

Let $a \in A$.

Let $\epsilon \in \R_{>0}$.

Let $\delta = \epsilon$.

Then:

\(\ds \map d {x, a}\) \(<\) \(\ds \delta\)
\(\ds \leadsto \ \ \) \(\ds \map d {\map {I_A} x, \map {I_A} a}\) \(=\) \(\ds \map d {x, a}\)
\(\ds \) \(<\) \(\ds \delta\)
\(\ds \) \(=\) \(\ds \epsilon\)

$\blacksquare$


Sources