Warm Ups. AP CALCULUS 2.4 Continuity Obj: identify the types of discontinuity.

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Presentation transcript:

Warm Ups

AP CALCULUS 2.4 Continuity Obj: identify the types of discontinuity

Review Complete the table to show. Use the graph to find a) and b).

Review Find the limits.

Continuity 3 Types of Discontinuity Undefined at cLimit does not exist Limit exists at c, but at c is not equal to f(c).

Definition of Continuity A function is continuous at c if the following conditions are met. 1.f(c) is defined. 2. exists. 3. = f(c). A function is continuous on an open interval (a, b) if it is continuous at each point in the interval.

Discontinuity Discontinuity is either removable or nonremovable. A function f has removable discontinuity if it can be made continuous by defining f at c. Discuss the continuity of the following functions. f(x) = [x]f(x) = 1/x-2f(x) = f(x) =

One Sided Limits Find the limits. A limit at c exists iff the limit is equal from both sides.

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 and The function f is continuous from the right at a and continuous from the left at b.

Properties of Continuity If f and g are continuous at x = c, then the following functions are also continuous at c. 1.bf 2.f + g 3.fg 4.f/gif g(c) ≠ 0

Intermediate Value Theorem If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k. Assignment: page 98 1 – 45 odd, 83, 87, 91, 93