A function defined on a subset is continuous (in full, continuous on ), if it is continuous at each point . Continuity is equivalent to the requirement that the preimage of any open set is open in (i.e., is an intersection between and an open subspace ).
- Continuity at a point is a local property. Automatic continuity at isolated points of .
- Examples of discontinuity points.
- Continuity and arithmetic operations. Continuity of the composition. Continuity of elementary functions.
- Dirichlet function: the ugly beast. Further pathologies.
- Continuity and global properties:
- Continuity and boundedness
- Continuity and existence of roots
- Continuity and monotonicity
- Continuity and functional equations , , , .