Continuity on Open & Closed Intervals Objective: Be able to describe where a function is continuous and classify any discontinuities as removable or non-removable.

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Presentation transcript:

Continuity on Open & Closed Intervals Objective: Be able to describe where a function is continuous and classify any discontinuities as removable or non-removable.

Definition of Continuity Continuity at a point: A function f is continuous at c if the following three conditions are met. 1.f(c) is defined. 2. exists. 3. Continuity on an open interval: A function is 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 line (-∞,∞) is everywhere continuous.

Removable Vs. Non-Removable Discontinuities If a function f is defined on an interval and f is not continuous at c, then f is said to have a discontinuity at c. A discontinuity is removable if f can be made continuous by appropriately defining (or redefining) f(c). Otherwise the discontinuity is non-removable. Examples:

Examples Discuss the continuity of each function.

Definition of Continuity on a Closed Interval A function f is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and The function f is continuous from the right at a and continuous from the left at b.

Examples Discuss the continuity of each function.

Examples Discuss the continuity of each function.

Properties of Continuity

Examples Discuss the continuity of each function.

Example: Thinking a different way. Find a such that the function is continuous on the entire real line.