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 ) &gt; 0 on R or f ′ ( x ) &lt; 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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My textbook Elementary Classical Analysis claims that by Darboux&#039;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  )  &amp;gt;  0 on       R   or       f    ′    (  x  )  &amp;lt;  0 on       R  ).The claim is definitely true if       f    ′    s domain were instead a closed interval, but since       R   is open, I don&#039;t understand why Marsden&#039;s claim should be true.

Kali Galloway · Accepted Answer

Argue by contradiction. If   f is not strictly monotonic, then,   f&#039; 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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