Definition of Continuity

AP Calculus · Axiom Academy

LESSON Definition of Continuity When a function has no breaks, jumps, or holes A function is continuous at a point if you can draw its graph without lifting your pencil. More formally, the function's value equals its limit at that point. A function is continuous on an interval if it's continuous at every point in that interval. When continuity fails, you get discontinuities: Removable: There's a hole, but the function could be continuous if we redefined one point Jump: Left and right limits exist but are different Infinite: Function approaches ±∞ (vertical asymptote) Oscillating: Function bounces wildly Continuity is essential for many calculus theorems: The Intermediate Value Theorem requires continuity Differentiability requires continuity Many limit laws assume continuity

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