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Chapter 2.5. Continuity and One-Sided Limits. Continuous – the graph of f(x) is uninterrupted—that is, unbroken—no holes, jumps, or gaps. Definition of Continuity. Continuity at a Point : A function is called continuous at c if the following 3 conditions are met.
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Chapter 2.5 Continuity and One-Sided Limits
Continuous – the graph of f(x) is uninterrupted—that is, unbroken—no holes, jumps, or gaps Definition of Continuity • Continuity at a Point: A function is called continuous at c if the following 3 conditions are met. • 1) is defined2) exists3) • Continuity on an Open Interval: A function is called continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real lineis called everywhere continuous
A function is said to be discontinuous at c if is defined on an open interval containing c (except possibly at c) and f is not continuous at c.
Two Categories of Discontinuities • 1) Removable: A discontinuity at x=c is removable if can be made continuous by appropriately defining (or redefining) at x=c. • 2) Nonremovable: For example, if the graph is broken or has gaps
The Intermediate Value Thm • If is continuous on the closed interval [a,b] and k is any number between and then there is at least one number c in (a,b) such that . • Note: If is continuous on [a,b] and and differ in sign, then the IVT guarantees the existence of at least one zero of in the closed interval [a,b].
