Continuity at a Point

Definition

Continuity at a point $x = c$ ensures that a function’s behavior is predictable and unbroken at that location, with no gaps, jumps, or holes in its graph.

Why It Matters

It guarantees that small changes in input lead to small changes in output, which is a prerequisite for any stable engineering design.

Core Concepts

• The Three-Part Test: $f$ is continuous at $c$ if:
  1. $f(c)$ is defined (the point exists).
  2. $\lim_{x \to c} f(x)$ exists (the approach is consistent).
  3. $\lim_{x \to c} f(x) = f(c)$ (the destination matches the value).
  • How to read: “The limit as x approaches c of f of x equals f of c.”
  • Meaning / when to use: All three conditions must hold—no hole, no jump, no isolated point. Prerequisite for EVT, IVT, and differentiability.
• One-Sided Continuity: Right-continuity ($\lim_{x \to c^+} f(x) = f(c)$) and left-continuity ($\lim_{x \to c^-} f(x) = f(c)$) allow for continuity at domain endpoints.
  • How to read: “The limit as x approaches c from the right equals f of c, and the limit as x approaches c from the left equals f of c.”
  • Meaning: One-sided continuity—continuity from one direction only; needed at interval endpoints.

Connected Concepts

• Derivative at a Point
• Slope
• Lines
• Distance from a Point to a Plane