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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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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
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
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
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
Example 1:

Show that is everywhere continuous.


Example 2
Example 2:

Where is continuous? Where is it not?


Removing a discontinuity
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)?


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