Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

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

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

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
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.

share|improve this answer
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

Not the answer you're looking for? Browse other questions tagged or ask your own question.