138 Continuity
A property of a mapping under which small changes in the input produce small changes in the value that is used to ensure limits and calculus behave well.
Definition (Continuity) A property of a map between open subsets, \(f: X \rightarrow Y\), where the following conditions apply:
- The preimage of every open subset of \(Y\) is an open subset of \(X\).
where
- \(X, Y \subseteq \mathbb{R}^n\)
- \(f^{-1}(U) = \{x \in X : f(x) \in U\}\)
- \(f^{-1} : \mathcal{P}(Y) \rightarrow \mathcal{P}(X)\) is the preimage map from subsets of \(Y\) to subsets of \(X\).
- \(U \subseteq Y\)
- \(\mathcal{P}(X)\) is the power set of \(X\)
138.1 Elementary Example
138.1.1 Simple
A map is continuous when the preimage of every open set is open. On finite spaces, check each open set in the codomain.
\[ X = Y = \{ 1,\ 2,\ 3 \} \]
\[ \tau = \{ \emptyset,\ \{1\},\ \{1,2\},\ X \} \]
\[ f(1)=1,\ f(2)=1,\ f(3)=2 \]
\[ f^{-1}(\{1\}) = \{ 1,\ 2 \} \in \tau \]
where
- \(\tau\) is the topology on \(X\) and on \(Y\).
- \(f^{-1}(\{1\})\) is open, as required.