continu.ax

Definition

A function f is continuous at a point c if the limit of f(x) as x approaches c equals f(c). No jumps, no gaps, no breaks. The value exists, the limit exists, and they agree.

lim_x→c f(x) = f(c)

Intermediate Value Theorem

If f is continuous on [a, b] and k is any value between f(a) and f(b), then there exists some c in (a, b) such that f(c) = k. Continuity guarantees that all intermediate values are achieved.

∀ k ∈ [f(a), f(b)], ∃ c ∈ (a, b) : f(c) = k

Uniform Continuity

A stronger condition than pointwise continuity. For every epsilon, a single delta works uniformly across the entire domain. The rate of change is bounded everywhere.

∀ ε > 0, ∃ δ > 0 : |x − y| < δ ⇒ |f(x) − f(y)| < ε

Lipschitz Continuity

The distance between function values is bounded by a constant multiple of the distance between inputs. The function cannot change faster than a fixed rate.

|f(x) − f(y)| ≤ K · |x − y|

Density of the Continuum

Between any two real numbers, there exists another real number. The real line has no gaps. This density property is what makes the continuum continuous -- an unbroken, infinitely divisible axis stretching in both directions.