by Darboux's theorem (the intermediate value theorem for derivatives), if a function f:R->R has a nonzero derivative on R, then is f strictly monotonic (i.e., either f′(x)>0 on R or f′(x)<0 on R). The claim is definitely true if f′s domain were instead a closed interval, but since R is open, I don't understand why Marsden's claim should be true.

Stephanie Hunter 2022-07-18 Answered
My textbook Elementary Classical Analysis claims that by Darboux's theorem (the intermediate value theorem for derivatives), if a function f : R R has a nonzero derivative on R , then is f strictly monotonic (i.e., either f ( x ) > 0 on R or f ( x ) < 0 on R ).
The claim is definitely true if f s domain were instead a closed interval, but since R is open, I don't understand why Marsden's claim should be true.
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Answers (1)

Kali Galloway
Answered 2022-07-19 Author has 16 answers
Argue by contradiction. If f is not strictly monotonic, then, f' would take both a positive value and a negative value. There is a closed interval containing the points at which these values occur. So...
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