Continuity
Definition
f is continuous at a if: (1) f(a) is defined, (2) \(\lim_{x\to a}f(x)\) exists, (3) \(\lim_{x\to a}f(x)=f(a)\).
Types of Discontinuity
| Property | Statement |
|---|---|
| Removable | Limit exists but ≠ f(a) |
| Jump | Left and right limits differ |
| Infinite | Limit is ±∞ |
Examples
Example 1. Is f(x)=sin(x)/x continuous at x=0?
Solution. No (undefined), but the removable discontinuity is fixed by defining f(0)=1.
Deep Dive: Continuity
This section builds durable understanding of continuity in calculus through definition-first reasoning, theorem mapping, and error-checking workflows.
Use a two-pass method: first derive the structure symbolically, then validate with a concrete numerical or geometric test case.
Visual Intuition
Convert algebra into a diagram, graph, or dependency map before solving. Visual-first analysis reduces sign errors and makes assumptions explicit.
Practice Set
Practice A. Re-derive one key formula on this page from first principles and annotate each transformation.
Target. Your final line should include assumptions, derivation path, and a quick verification.
Practice B. Build an application scenario using continuity and solve it with both symbolic and numeric methods.
Target. Compare outputs and explain any approximation gap.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Editorial update: Reviewed on 2026-04-14 for notation consistency, conceptual clarity, and exercise quality.