calculus

Continuity

A function is continuous at a point if its value there equals the limit of its values as inputs approach the point — no jumps, holes, or asymptotes.

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A function ff is continuous at x=ax = a if three conditions hold:

  1. f(a)f(a) is defined,
  2. lim⁡x→af(x)\lim_{x \to a} f(x) exists, and
  3. lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a).

Intuitively: you can draw the graph through that point without lifting your pen. Common discontinuities are removable (a hole), jump (left and right limits differ), and infinite (vertical asymptote).

Continuity is the entry-level requirement for most calculus theorems. The Intermediate Value Theorem says continuous functions take every value between any two outputs. The Extreme Value Theorem guarantees continuous functions on a closed interval attain a maximum and minimum. Differentiability requires continuity, but continuity does not imply differentiability — ∣x∣|x| is continuous everywhere yet not differentiable at 00.