Continuity
Concept · Mathematics, stage 1: Learn the vocabulary
A function is continuous when small changes to the input produce only small changes to the output, so its graph has no jumps or breaks. The precise version says that for any tolerance on the output there is a tolerance on the input that guarantees it. Topology takes this further by throwing away distance entirely and keeping only which points count as near which: a continuous map is then one that never tears the space apart, though it may stretch or bend it.
Why it matters
It is the property that lets you reason about a whole curve from what happens near single points, and generalising it is how topology came to exist.
Part of
Sources
- Continuous function Wikipedia
- Topological space WikipediaShows what is left of continuity once distance is removed.
- Continuity and Infinitesimals Stanford Encyclopedia of Philosophy
