Definition:Continuous Mapping (Metric Space)/Space
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Definition
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.
Definition 1
$f$ is continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$ if and only if it is continuous at every point $x \in A_1$.
Definition 2
$f$ is continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$ if and only if:
- for every $U \subseteq A_2$ which is open in $M_2$, $f^{-1} \sqbrk U$ is open in $M_1$.
Also known as
A mapping which is continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$ can also be referred to as $\tuple {d_1, d_2}$-continuous.