Metric space. Function continuity
Function continuity between metric spaces
Let \( X \) and \( Y \) be metric spaces and let \( f: X \to Y \) be a function. We say that \( f \) is continuous iff:
For all \( x_0 \in X \) and for all \( \varepsilon > 0 \) there exists a [...] such that ([...] \( \implies \) [...]).
Independence of metrics
The idea of continuity appears to depend on the notion of metric/distance as it is formulated using ε-balls. However, the following lemma reveals that this is not really so—all that is needed is the notion of open (or closed) subsets.
Equivance to function continuity
Let \( X \) and \( Y \) be metric spaces and let \( f : X \to Y \) be a function. The the following three statements are equivalent:
- \( f \) is continuous;
- for all open \( U \subseteq Y \), [...];
- for all closed \( V \subseteq Y \), [...].
Informally: [2. can be rephrased as...].
Proof at the bottom of the back side.
This result motivates the definition of a topolgical space.