Types of Discontinuity

Definition

A discontinuity occurs at a point $x = c$ when a function fails the continuity test. These failures are categorized by the nature of the “break” in the function’s graph.

Why It Matters

Systems don’t just “stop”; they fail in specific, predictable patterns like jumps, holes, or unstable oscillations. Identifying the type of discontinuity is the first step in triage, providing a diagnostic manual for the exact mechanism of failure in a model or an engineering design.

Core Concepts

• Removable: $\lim_{x \to c} f(x)$ exists, but $f(c)$ is either undefined or doesn’t match the limit. A “hole” in the graph.

  • How to read: “The limit of f of x as x approaches c exists, but f of c is either undefined or not equal to that limit.”
  • Meaning: Nearby values agree on a target, but the point itself is wrong—often fixable by redefining $f(c)$.

• Jump: Left-hand and right-hand limits exist but are not equal ($\lim_{x \to c^-} \neq \lim_{x \to c^+}$).

  • How to read: “The limit of f of x as x approaches c from the left is not equal to the limit of f of x as x approaches c from the right.”
  • Meaning: The graph steps from one plateau to another—two different approaching values, so no single limit.

• Infinite: One or both one-sided limits are $\pm\infty$, creating a vertical asymptote.

  • How to read: “The limit of f of x as x approaches c from either side equals positive or negative infinity.”
  • Meaning: Outputs grow without bound near c—the graph shoots off vertically.

• Oscillating: The function fluctuates infinitely often between values as $x \to c$ (e.g., $\sin(1/x)$).

  • How to read: “The function sine of one divided by x has a limit that does not exist as x approaches zero.”
  • Meaning: No limit because the function wiggles faster and faster without settling on one value.

Connected Concepts

• Artificial Intelligence
• Systems Thinking
• Common Mathematical Function Types