The Intermediate Value Theorem states that, for a continuous function f : [ a , b

ddcon4r 2022-07-10 Answered
The Intermediate Value Theorem states that, for a continuous function f : [ a , b ] R , if f ( a ) < d < f ( b ), then there exists a c ( a , b ) such that f ( c ) = d .

I wonder if I change the hypothesis of f ( a ) < d < f ( b ) to f ( a ) > d > f ( b ), the result still holds. I believe so, since f assumes a fixed point f ( x ) = x in [ a , b ], so we would have c = d, although I'm not completely sure.

I need this result in order to prove the set X = { x [ a , b ] s . t . f | [ a , x ] is bounded }, with a continuous f, is not empty.
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Answers (1)

Aryanna Caldwell
Answered 2022-07-11 Author has 11 answers
I don't know how you're getting that f has a fixed point, or how that would imply that c = d. You seem to be confusing inputs and outputs of f: we know nothing at all about how f ( a ) and f ( b ) are related to a and b.

However, you can instead just apply the Intermediate Value Theorem to the function g ( x ) = f ( x ), which satisfies g ( a ) < d < g ( b ), to get some c such that g ( c ) = d, so f ( c ) = d.

(It is not clear to me what your question has to do with proving that the set X is nonempty, though.)

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