Continuous Function: Is There an Open Interval Where f is Monotone?

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A continuous function f on R may not necessarily have an open interval where it is monotone. While non-constant functions can exhibit monotonicity in certain intervals, continuity alone does not guarantee this behavior. The Weierstrass function is cited as an example of a continuous function that is nowhere differentiable, highlighting the distinction between continuity and monotonicity. To determine monotonicity, knowledge of the sign of the derivative f '(x) in a neighborhood is essential. Therefore, continuity does not imply the existence of a monotone interval.
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Homework Statement



Let f be continuous on R. Is there an open interval on which f is monotone?

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The Attempt at a Solution



I think there is such interval for non constant function but I am really not sure.
 
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You might want to check out the "Weierstrasse function" which is continuous for all x but differentiable nowhere.
 
In general, no.

Continuity doesn't tell you how the fuction approach it's points. If you knew that f '(x) was either negative or postive around some neighbour of your point then you can say if it is monotone or not.
 

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