# a) Identify the following discontinuities. b) Find a function that could describe the graph. - PowerPoint PPT Presentation

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Opener :. a) Identify the following discontinuities. b) Find a function that could describe the graph. c) Name an interval on which all of these functions are continuous. M439—Calculus A. Section 2.3: Continuity Continued. Intermediate Value Theorem. f(b). f(a). b. a.

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a) Identify the following discontinuities. b) Find a function that could describe the graph.

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#### Presentation Transcript

Opener:

a) Identify the following discontinuities.

b) Find a function that could describe the graph.

c) Name an interval on which all of these functions are continuous

### M439—Calculus A

Section 2.3: Continuity Continued

Intermediate Value Theorem

f(b)

f(a)

b

a

If y = f(x) is continuous on a closed interval [a,b], then it takes on every value between f(a) and f(b) (and sometimes a couple extras)

Connectivity:

f(b)

f(a)

b

a

The intermediate value theorem implies that a continuous function will have no breaks (no jumps, holes, or limits whose values approach infinity).

### Car Application

If your car is traveling at some speed, it must have gone every speed between 0 mph and your current speed during your drive.

### Properties of Continuous Functions:

• If two functions , f and g, are continuous at x = c, then most combinations of f and g are continuous at x = c:

1) Sums: f + g

2) Differences: f – g

3) Products: f • g

4) Scalar Multiples: k • f (for any real number k)

5) Quotients: f/g (as long as g(c) is nonzero)

6) Compositions: f о g

### Example 1:

Show that is everywhere continuous.

### Example 2:

Where is continuous? Where is it not?

### Removing a Discontinuity

We notice that g(x) has a “removable” discontinuity.

What does this mean? Which other discontinuities are NOT removable?

How do we remove it?

How do we extend the function to a new function, h(x)?