Most of the techniques of calculus require that functions be continuous. A function is continuous if...

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Most of the techniques of calculus require that functions be continuous . A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function 1 2 3 4 1 2

Transcript of Most of the techniques of calculus require that functions be continuous. A function is continuous if...

Page 1: Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your.

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.

A function is continuous at a point if the limit is the same as the value of the function.

This function has discontinuities at x=1 and x=2.

It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

1 2 3 4

1

2

Page 2: Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your.

jump infinite oscillating

Essential Discontinuities:

Removable Discontinuities:

(You can fill the hole.)

Page 3: Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your.

Removing a discontinuity:

has a discontinuity at .

Write an extended function that is continuous at .

Note: There is another discontinuity at that can not be removed.

Page 4: Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your.

Removing a discontinuity:

Note: There is another discontinuity at that can not be removed.

Page 5: Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your.

Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.

Also: Composites of continuous functions are continuous.

examples:

Page 6: Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your.

Intermediate Value Theorem

If a function is continuous between a and b, then it takes

on every value between and .

Because the function is continuous, it must take on every y value between and .

Page 7: Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your.

Example 5: Is any real number exactly one less than its cube?

(Note that this doesn’t ask what the number is, only if it exists.)

Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5.Therefore there must be at least one solution between 1 and 2.

Use your calculator to find an approximate solution.

F2 1: solve

Page 8: Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your.

This example was graphed on the classic TI-89. You can not change the resolution on the Titanium Edition.

Graphing calculators can sometimes make non-continuous functions appear continuous.

Graph:

CATALOG F floor(

Note resolution.

The calculator “connects the dots” which covers up the discontinuities.

Page 9: Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your.

Graphing calculators can make non-continuous functions appear continuous.

Graph:

CATALOG F floor(

GRAPH

The open and closed circles do not show, but we can see the discontinuities. p

If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct floor graph.


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