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Continuity Curves without gaps?

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Continuity Curves without gaps?. Animation (infinite length). Continuity. Definition A function f is continuous at a number a if lim f(x) = f(a). x -> a. 1. f (a) is defined. 2. lim f(x) exists. x -> a. 3.lim f(x) = f(a). x -> a.

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Presentation Transcript
slide1
Continuity

Curves without gaps?

Animation

(infinite length)

slide2
Continuity

DefinitionA function f is continuous at a number aif

lim f(x) = f(a).

x -> a

1. f (a) is defined.

2. lim f(x) exists.

x -> a

3.lim f(x) = f(a).

x -> a

slide3
Continuity

1. f (a) is defined.

2. lim f(x) exists.

x -> a

3.lim f(x) = f(a).

x -> a

Animation

sin (1/x)

slide4
If f is not continuous at a , we say f is discontinuous at a, or f has a discontinuity at a .

Example Where is the function

f(x)=(x 2 – x – 2)/(x – 2) discontinuous?

slide5
DefinitionA function f is continuous from the right at a number a if

lim f(x) = f(a),

and f is continuous from the left a if

lim f(x) = f(a).

x -> a +

x -> a-

slide6
DefinitionA function f is continuous on an interval if it is continuous at every number in the interval.

At an endpoint of the interval we understand continuous to mean continuous from the right or continuous from the left.

slide7
Example Use the definition of continuity and the properties of limits to show that the function

f (x) = x 16 –x2is continuous on the interval [-4, 4].

_____

slide8
Theorem If f and g are continuous at a and c is a constant, then the following functions are also continuous at a:

1. f + g 2. f – g 3. c f

4. f g 5. ( f / g) if g(a) is not

equal to 0.

slide9
Theorem
  • Polynomials are continuous everywhere; that is continuous on R = (-, ).
  • Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.
slide10
TheoremThe following types of functions are continuous at every number in their domains:

-polynomials -rational functions -root functions -exponential functions -trigonometricfunctions -inverse trigonometric functions -logarithmic functions.

slide11
Example Evaluate

lim arctan ((x 2- 4) / (3x 2 – 6x)).

x -> 2

slide12
Theorem If f is continuous at b and lim g(x) = b, then, lim f(g(x)) = f(b).

In other words,

lim f(g(x)) = f(lim g(x)).

x -> a

x -> a

x -> a

x -> a

slide13
Theorem If g is continuous at a and f is continuous at g(a), then

(f o g)(x) = f(g(x)) is continuous at a.

slide14
The Intermediate Value Theorem

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

slide15
Example Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

ln x = e –x , (1,2).

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