Warm up Evaluate the limit Section 2.4 Continuity SWBATSWBAT –Define continuity and its types.

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Presentation transcript:

Warm up Evaluate the limit

Section 2.4 Continuity SWBATSWBAT –Define continuity and its types

Conceptual continuity

2.4 Continuity This implies :This implies : 1.f(a) is defined 2.f(x) has a limit as x approaches a 3.This limit is actually equal to f(a).

Definition (cont’d)

Types of discontinuity Removable Discontinuity: “A hole in the graph” (You can algebraically REMOVE the discontinuity)

Types of discontinuity (cont’d) Infinite discontinuity: Where the graph approaches an asymptote It can not be algebraically removed

jump discontinuity the function “jumps” from one value to another.

Example Where are each of the following functions discontinuous, and describe the type of discontinuityWhere are each of the following functions discontinuous, and describe the type of discontinuity

One-Sided Continuity Continuity can occur from just one side:Continuity can occur from just one side:

Continuity on an Interval So far continuity has been defined to occur (or not) one point at a time.So far continuity has been defined to occur (or not) one point at a time. We can also consider continuity over an entire interval at a time:We can also consider continuity over an entire interval at a time: Continuous on an Interval: it is continuous at every point on that interval.Continuous on an Interval: it is continuous at every point on that interval.

Polynomials and Rational Functions Write the interval where this function is continuous.Write the interval where this function is continuous.

Types of Continuous Function We can prove the following theorem:We can prove the following theorem: This means that most of the functions encountered in calculus are continuous wherever defined.This means that most of the functions encountered in calculus are continuous wherever defined.

1. Lim f(x) x  Lim f(x) x  Lim f(x) x  -  4. Lim f(x) x  Lim f(x) x  Lim f(x) x  0 7. f(2)8. f(-2)

Assignment 8 p oddp odd Quiz tomorrow –Quiz tomorrow – 2.1 through 2.4 Continuity