# Continuity

**Type:** Concept  
**Domain:** Analysis  
**Codex URL:** /mathematics/analysis/continuity/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A function with no sudden jumps or breaks. Defined rigorously via limits.

## Definition
f continuous at a: f(a) defined, lim(x→a)f(x) exists, and lim(x→a)f(x)=f(a).

## Types of discontinuity
Removable (patchable hole), jump (mismatched one-sided limits), infinite (blows up).

## Intermediate Value Theorem
If f continuous on [a,b], f takes every value between f(a) and f(b).

## Extreme Value Theorem
Continuous f on closed bounded [a,b] attains both max and min.

## Continuity vs differentiability
Differentiable ⟹ continuous, but not conversely (|x| at x=0).

## Uniform continuity
Single δ works for all points — automatic on closed bounded intervals (Heine-Cantor).

## Topological definition
f continuous iff preimage of every open set is open — generalizes beyond metric spaces.

## Weierstrass function
Continuous everywhere, differentiable nowhere (1872) — shocked 19th-century mathematicians.

## Sources
### Tier 1
- Rudin, W. (1976). *Principles of Mathematical Analysis*. 3rd ed.
- Munkres, J.R. (2000). *Topology*. 2nd ed.
### Tier 2
- Spivak, M. (2008). *Calculus*. 4th ed.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*