I am asked to find an example of a discontinuous function f : [ 0 , 1 ] &#x

Dayami Rose 2022-06-26 Answered
I am asked to find an example of a discontinuous function f : [ 0 , 1 ] R where the intermediate value theorem fails. I went over the intermediate value theorem today

Let f : [ a , b ] R be a continuous function. Suppose that there exists a y such that f ( a ) < y < f ( b ) or f ( a ) > y > f ( b ) . Then there exists a     c [ a , b ] such that f ( c ) = y.

I understand the theory behind it, however, we did not go over many example of how to use it to solve such problems so I do not really know where to begin
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Answers (1)

grcalia1
Answered 2022-06-27 Author has 23 answers
Look for a function with a jump discontinuity.
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Suppose that f is a continuous function and that f ( 1 ) = f ( 1 ) = 0. Show that there is c ( 1 , 1 ) such that
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By the way, the 'similar question' I mentioned above is as follow:

Suppose that f is a continuous function and that f ( 0 ) = 1 and f ( 1 ) = 2. Show that there is c ( 0 , 1 ) such that
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Suppose that f : [ 0 , 1 ] [ 0 , 2 ] is continuous. Use the Intermediate Value Theorem to prove that their exists c [ 0 , 1 ] such that:
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I know that when we have the condition were f : [ a , b ] [ a , b ], the method to prove that c exits, is the same method you would use to prove the fixed point theorem.

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