Introduction to Continuity

Business Calculus · Axiom Academy

LESSON Introduction to Continuity Understanding when functions behave "nicely" without breaks or jumps Intuitively, a function is continuous if you can draw its graph without lifting your pen from the paper. There are no gaps, jumps, or holes. In business terms, continuity means smooth, predictable behavior. A continuous cost function doesn't suddenly jump from 100 to 500. A continuous demand curve doesn't have missing price points. Most calculus tools (derivatives, integrals, optimization) require functions to be continuous. Understanding continuity tells you when these powerful techniques can be applied! The Three Conditions for Continuity A function f is continuous at a point x = a if three conditions are met: Definition of Continuity at a Point f(a) exists — the function is defined at a _ x a f(x) exists — the limit exists _ x a f(x) = f(a) — the limit equals the function value All three must be true! If any condition fails, the function is discontinuous at that point. Is f(x) = x^2 + 3x - 1 continuous at x = 2? f(2) = 4 + 6 - 1 = 9 ✓ (exists) _ x 2 (x^2 + 3x - 1) = 9 ✓ (limit exists) Limit = f(2) = 9 ✓ (they're equal) Yes! All polynomials are continuous everywhere. When continuity fails, it fails in predictable ways. There are three types of discontinuities: Limit exists, but f(a) is missing or wrong. Can be "fixed." Left and right limits exist but differ. Can't be fixed. Function goes to ±∞. Vertical asymptote.

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