Proving the existence of a non-monotone continuous function defined on [0,1]
Let be the sequence of intervals of [0,1] with rational endpoints, and for every let . Prove that for every , is closed and nowhere dense in . Deduce that there are continuous functions in the interval [0,1] which aren't monotone in any subinterval.
For a given n, can be expressed as where and are the sets of monotonically increasing functions and monotonically decreasing functions in respectively. I am having problems trying to prove that these two sets are closed. I mean, take such that there is with . How can I prove ?. Suppose I could prove this, then I have to show that . This means that for every and every , there is such that g is not monotone. Again, I am stuck. If I could solve this two points, it's not difficult to check the hypothesis and apply the Baire category theorem to prove the last statement.