Continuity
A function is continuous where small changes in input produce only small changes in output — no jumps or breaks.
At a glance
Key signals
0% cross fields · reaches 1 more
- Mathematical Analysis
- Mathematics
- Explanation
- Examples
- Misconception
- Sourced relations
- Attribution
Dependencies
What this concept builds on and what it makes possible — derived from the atlas’s dependency, causal and structural relations, not from every related edge.
Foundations · builds on
Continuitydepends onLimitEstablished
Continuity is defined by limits.
Mechanism: A function is continuous at a point when its value there equals the limit approaching that point.
Continuitydepends onLimitEstablished
Continuity depends on Limit.
Mechanism: A function is continuous at a point when its limit there equals its value.
Enables · leads to
Derivativedepends onContinuityEstablished
Differentiability requires continuity.
Mechanism: If a function has a derivative at a point it must be continuous there; the converse fails (continuity does not imply differentiability).
DerivativerequiresContinuityEstablished
Derivative requires Continuity.
System context
Uniform continuityis aContinuityEstablished
Uniform continuity is a kind of Continuity.
Structural role & consequence
Interpreted from the current atlas graph — what the connections mean, not just how many there are.
Currently dark in the atlas: no key date stored · 5 of 5 of its relations lack claim-level evidence.
atlas representation · Describes the current Thinking OS representation, not the state of the world.
Builds on 2 foundations (requires / depends-on / derived-from / emerges-from).
structural · Structural graph analysis — not a claim of importance, causation or history.
Structural neighbourhood: 4 → 17 → 34 concepts reachable within 3 hops.
structural · Structural reach — being reachable is not the same as being understood.
All 4 of its relationships stay within its own discipline — a field-specific concept in the current atlas.
structural · Structural graph analysis — not a claim of importance, causation or history.
cross-field
4 within-field, 0 cross-field
Strengths & constraints
Constraints
- Evidence coverage currently thin in the atlas — few of its relationships carry claim-level evidence. atlas representation
- No dated history stored — the atlas records no key date for this concept. atlas representation
Conditions
- Its dependency reading rests on 2 foundation relations. structural
- Read structurally — most of its relationships carry no external evidence yet, so claims here are graph-derived. structural
Dependency radial
What this concept builds on (left) and what it makes possible (right) — derived from dependency and causal relations.
What builds on this
1 concept build on this directly, 4 in total, across 3 disciplines.
Structural downstream reach along dependency edges — not a claim of historical necessity.
Seen through each discipline
How this concept sits in each of its fields — derived from its real connections in the graph, not asserted.
Through this lens it connects to Derivative, Derivative, Limit and Uniform continuity.
Through this lens it connects to Limit, Derivative and Derivative.
Related ideas to explore
Concepts that look related but are not yet connected here — candidates for a connection to reason about, not established links.
- Connects 4 other ideas across 2 disciplines.
- Most of its connections are of the “Dependency” kind.
Derived from the graph’s real structure — observations, not a score.
Sources
- On Darboux continuity and continuity (1981) verified
- Continuity (2006) verified
- The Metaphysics of Continuity (2023) verified
- Continuity and Uniform Continuity (2010) verified