3 2 continuity
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3.2: Continuity. Objectives: To determine whether a function is continuous Determine points of discontinuity Determine types of discontinuity Apply the Intermediate Value Theorem. CONTINUITY AT A POINT—No holes, jumps or gaps!!. A function is continuous at a point c if:

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3.2: Continuity

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3 2 continuity

3.2: Continuity

Objectives:

To determine whether a function is continuous

Determine points of discontinuity

Determine types of discontinuity

Apply the Intermediate Value Theorem


Continuity at a point no holes jumps or gaps

CONTINUITY AT A POINT—No holes, jumps or gaps!!

A function is continuous at a point c if:

1.) f(c) is defined

2.) exists

3.) = f(c)

A FUNCTION NEED NOT BE CONTINUOUS OVER ALL REALS TO BE A CONTINUOUS FUNCTION


3 2 continuity

Does f(2) exist?

Does exist?

Does exist?

Is f(x)continuous at x = 2?

Do the same for x= 1, 3, and 4.


Removable discontinuity hole in graph

Removable discontinuity(hole in graph)

  • Limit exists at c but f(c)≠ the limit

  • Can be fixed. Set f(c) =

  • This is called a continuous extension


3 2 continuity

Example: Find the values of x where the function is discontinuous. State if it is a removable discontinuity. If so, fix it.


Other types of discontinuity

OTHER TYPES OF DISCONTINUITY

  • JUMP: ( RHL ≠ LHL)

  • INFINITE:

  • OSCILLATING


Continuity on a closed interval

Continuity on a closed interval

A function is continuous on a closed interval [a,b] if:

  • It is continuous on the open interval (a,b)

  • It is continuous from the right at x=a:

  • It is continuous from the left at x=b:


Example

example

IT IS CONTINUOUS ON ITS DOMAIN. But discontinuous on x values not in the domain.


Key functions and where they are continuous

Key Functions and where they are continuous


3 2 continuity

A continuous function is one that is continuous at every point in its domain. It need not be continuous on all reals.

Where are the functions discontinuous? If it is removable discontinuity, fix it!!


For piecewise functions

For piecewise functions…

  • Check to make sure each “piece” is continuous

  • Check the x values where it changes functions. Remember, the following must be true to be continuous at x:


Where are the functions discontinuous fix removable

Where are the functions discontinuous? Fix removable!


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