CONTINUITY The man-in-the-street understanding of a continuous process is something that proceeds...
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![Page 1: CONTINUITY The man-in-the-street understanding of a continuous process is something that proceeds smoothly, without breaks or interruptions. Consequently,](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d2e5503460f94a05e62/html5/thumbnails/1.jpg)
CONTINUITYThe man-in-the-street understanding of a
continuous process is something that proceeds
smoothly, without breaks or interruptions.
Consequently, for a function to be
called “continuous”, we would expect its graph to
be a smooth line, without breaks or interruptions.
![Page 2: CONTINUITY The man-in-the-street understanding of a continuous process is something that proceeds smoothly, without breaks or interruptions. Consequently,](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d2e5503460f94a05e62/html5/thumbnails/2.jpg)
Let’s look at some graphs we would definitely not call continuous ; the best way to define “day” is
to think of “night”, “happy” is meaningless unless
you know “sad”, most concepts are better understood via their opposites !
Here is a function whose graph you would definitely not call continuous, it jumps at every integer!
The formal definition of is
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(The notation is somewhat different from the textbook’s, it means the greatest integer ≤ )Here is the graph
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There’s a break at every integer! What’s the trouble? Here is another
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A break at 3 again! Two more graphs.
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A hole at 2 !
On the right the hole has been incorrectly filled.The next example is the messiest.
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![Page 8: CONTINUITY The man-in-the-street understanding of a continuous process is something that proceeds smoothly, without breaks or interruptions. Consequently,](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d2e5503460f94a05e62/html5/thumbnails/8.jpg)
Talk about not smooth! A little better:
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What do these pictures tell us about our intuitive notion of a continuous graph?
There should be:
No holes
No jumps
No uncertainties.
To a mathematician these mean:
![Page 10: CONTINUITY The man-in-the-street understanding of a continuous process is something that proceeds smoothly, without breaks or interruptions. Consequently,](https://reader035.fdocuments.in/reader035/viewer/2022062516/56649d2e5503460f94a05e62/html5/thumbnails/10.jpg)
(the order is mixed up.)
These three are condensed in:
And formally:
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CONTINUITY AT
Definition. The function is said tobe continuous at if
(all three previous conditions are assured by this statement.)
Now by application of the three statements
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No. 1 If , where and
are polynomials, and then
we get that
every rational function is continuous at every point where it is defined.
No. 2 (usual caveat about n) gives us that
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radicals of continuous functions are continuous wherever the are defined.Finally, fromNo.3We get thatAll trigonometric functions are continuous wherever they are defined.Finally, if and are continuous atthen so are , , and if
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What about the composition ?A look at this picture tells us that
If is continuous at and is continuous
at then is continuous at
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Intermediate Value Theorem
Probably the most important (useful) property of continuous functions is the following
Theorem. If is continuous at every , then for every number
between and there is at least one such that
A picture will help:
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Here is the situation:
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You can’t join the two red dots “continuously without crossing the blue dotted line.