Continuity Definition Continuity A function is continuous at

  • Slides: 11
Download presentation
Continuity

Continuity

Definition: Continuity A function is continuous at a number a if That is, 1.

Definition: Continuity A function is continuous at a number a if That is, 1. f(a) is defined 2. exists 3.

Definition: One Sided Continuity A function f is continuous from the right at a

Definition: One Sided Continuity A function f is continuous from the right at a number a if and f is continuous from the left at a if

Definition: Continuity On An Interval A function f is continuous on an interval if

Definition: Continuity On An Interval A function f is continuous on an interval if it is continuous at every number in the interval. (If f is defined on one side of an endpoint of the interval, we understand continuous at the endpoints to mean continuous from the right or continuous from the left).

Theorem If f and g are continuous at a and c is a constant,

Theorem If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: 1. 2. 3. 4. 5. f+g f–g cf fg f / g if g(a) 0

Theorem (a) Any polynomial is continuous everywhere; that is, it is continuous on =

Theorem (a) Any polynomial is continuous everywhere; that is, it is continuous on = (-∞, ∞). (b) Any rational function is continuous whenever it is defined; that is, it is continuous on its domain.

Theorem Any of the following types of functions are continuous at every number in

Theorem Any of the following types of functions are continuous at every number in their domain: Polynomials; Rational Functions, Root Functions; Trigonometric Functions; Inverse Trigonometric Functions; Exponential Functions; and Logarithmic Functions.

Theorem If f is continuous at b and. In other words, , then

Theorem If f is continuous at b and. In other words, , then

Theorem If g is continuous at a and f is continuous at g(a), then

Theorem If g is continuous at a and f is continuous at g(a), then the composite function f(g(x)) is continuous at a.

The Intermediate Value Theorem n Suppose that f is continuous on the f(a) closed

The Intermediate Value Theorem n Suppose that f is continuous on the f(a) closed interval [a, b] f(c)=N and let N be any f(b) number between f(a) and f(b). Then there exists a number c in (a, b) such that f(c) = N. f a c b

Example • Use the Intermediate Value Theorem to show that there is a root

Example • Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

Present Continuous Past Continuous Future Continuous Present ContinuousPresent Continuous Past Continuous Future Continuous Present Continuous
BUSINESS CONTINUITY Business continuity definition Business Continuity ManagementBUSINESS CONTINUITY Business continuity definition Business Continuity Management
A 4 CONTINUITY Investigating Continuity Continuous Function AA 4 CONTINUITY Investigating Continuity Continuous Function A
Function Function Declaration Function Definition Calling a FunctionFunction Function Declaration Function Definition Calling a Function
CONTINUITY TASKS What is continuity Continuity is theCONTINUITY TASKS What is continuity Continuity is the
PAST CONTINUOUS Past Continuous Past Simple Past ContinuousPAST CONTINUOUS Past Continuous Past Simple Past Continuous
Continuous Distributions Continuous random variables For a continuousContinuous Distributions Continuous random variables For a continuous
CONTINUOUS IMPROVEMENT Co je Continuous Improvement Continuous ImprovementCONTINUOUS IMPROVEMENT Co je Continuous Improvement Continuous Improvement
English Grammar Tenses Continuous Continuous Present continuous PastEnglish Grammar Tenses Continuous Continuous Present continuous Past
Continuous Random Variables Discrete Vs Continuous Discrete ContinuousContinuous Random Variables Discrete Vs Continuous Discrete Continuous