Given two topological spaces \(X,Y\). A function \(f\colon X \to Y\) is called continuous if for every open \(O\subset Y\) the inverse image \(f^{-1}(O)\) is open.
Examples
Remarks
- We can replace open sets with closed sets in the definition, since \(f^{-1}(X\setminus A)=Y\setminus f^{-1}(A)\).
- We denote the space of all continuous functions on \(X\) with \(C(X)\).
Characterizations Link to heading
Sequences of continuous functions Link to heading
Restrictions and Expansions Link to heading
See also Link to heading
- homeomorphism
- open function
- closed function
- gluing lemma
- intermediate value theorem
- extreme value theorem