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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

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

Does f(2) exist?

Does exist?

Does exist?

Is f(x)continuous at x = 2?

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

- Limit exists at c but f(c)≠ the limit
- Can be fixed. Set f(c) =
- This is called a continuous extension

- JUMP: ( RHL ≠ LHL)
- INFINITE:
- OSCILLATING

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:

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

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

- 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: