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Please give an example of a continuous function f:[a,b]->R (a,b are real nos.), such that there is no open subinterval of [a,b],where f is monotone. Here, the constant function is considered monotone.

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closed as off topic by Jacques Carette, fedja, George Lowther, Alain Valette, Bill Johnson Dec 18 '11 at 16:54

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To be clear, every function monotone on an open interval is differentiable almost everywhere there. So any example of a continuous, nowhere-differentiable function will do. –  Michael Greinecker Dec 18 '11 at 16:10
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If you want an everywhere differentiable function, see artofproblemsolving.com/Forum/viewtopic.php?f=70&t=2822 In any case, this is neither a research level question, nor something amusing enough to make one tempted to ignore the proclaimed MO rules. So, I'm voting to close. –  fedja Dec 18 '11 at 16:27

1 Answer 1

Enumerate the rationals in [0,1] as q1,q2,... . Plot the points (q1,1) and (q2,1) to start with, followed by joining them by a straight line. Assume inductively that you have plotted the points upto some qn.Then, q(n+1) may lie at the extreme left or right of all the previously plotted qs, or lie strictly between some qk and ql (k,l<=n).In the 1st 2 cases, just join q(n+1) with the previous extreme point, after plotting q(n+1) arbitrarily(keeping some bound in mind). In the 3rd case plot(q(n+1),m), where m is any real within the bound, with m>max(y-coordinates of qk and ql). Then, rub the line joining qk,ql and draw 2 new lines joining qk,q(n+1) and ql. This is just a construction and you can obviously make it rigorous.

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Please don't answer non-research level questions like this - it encourages others to post inappropriate questions –  Anthony Quas Dec 18 '11 at 18:34
    
I completely agree with Anthony that the question is indeed a non-research level one, in fact, may be of a general home-work type too! –  Somabha Mukherjee Dec 19 '11 at 8:05

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