Proving the existence of a non-monotone continuous function defined on [0,1]

wstecznyg5 2022-07-22 Answered
Proving the existence of a non-monotone continuous function defined on [0,1]
Let ( I n ) n N be the sequence of intervals of [0,1] with rational endpoints, and for every n N   let E n = { f C [ 0 , 1 ] : f is monotone in I n }. Prove that for every n N , E n is closed and nowhere dense in ( C [ 0 , 1 ] , d ). Deduce that there are continuous functions in the interval [0,1] which aren't monotone in any subinterval.
For a given n, E n can be expressed as E n = E n E n where E n and E n are the sets of monotonically increasing functions and monotonically decreasing functions in E n respectively. I am having problems trying to prove that these two sets are closed. I mean, take f ( C [ 0 , 1 ] , d ) such that there is { f k } k N E n with f k f. How can I prove f E n ?. Suppose I could prove this, then I have to show that E n ¯ = E n = . This means that for every f E n and every r > 0, there is g B ( f , r ) 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.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Kali Galloway
Answered 2022-07-23 Author has 16 answers
Step 1
It is easy to show that E n is close. Just like what you did, let E n 1 be the subset of E n containing all functions increasing on I n . Let { f k } be a sequence in E n converging to f C [ 0 , 1 ]. Pick x , y I n , x < y. Then f n ( x ) f n ( y ) for all n. Take n gives f ( x ) f ( y ). Thus E n 1 is closed. Now E n is the union of two close set, so it is close.
To show that E n has empty interior, let f E n . Assume that f is inceasing (the case when f is decreasing can be done similarly). We perturb f a little bit to get a function h which is not monotone. Let Write I n = [ a , b ]. Let ϵ > 0 be arbitrary. Let c 1 , c 2 ( a , b ) such that c 1 < c 2 ,  and    f ( b ) ϵ < f ( c 2 ) f ( b ) .
c 1 , c 2 can be found as f is continuous. Let g be a continuous function on [0,1] with the properties:
| g | ϵ ,   g ( a ) = g ( b ) = ϵ ,   g ( c 1 ) = g ( c 2 ) = 0 ,   .
Step 2
Then h = f g satisfies d ( f , h ) ϵ and h ( a ) = f ( a ) ϵ < f ( a ) f ( c 1 ) = h ( c 1 ) and h ( c 2 ) = f ( c 2 ) > f ( b ) ϵ = h ( b ). Thus h is not monotone. As f, ϵ > 0 are arbitrary, E n contains no open set. Hence E n are nowhere dense. (The conclusion is quite interesting).

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-09-20
Let f ( x ) = ln ( x ) x . Find the critical values of f(x)
How would you find the critical value of this equation? So far... I have gotten that you find the derivative of f(x) which is f ( x ) = 1 x 3 / 2 ln ( x ) 2 x 3 / 2 ..
To find the critical values set f ( x ) = 0? But how would you factor out the derivative to find the critical values?
Also, after that how would you find the intervals of increase and decrease and relative extreme values of f(x).
And lastly the intervals of concavity and inflection points.
asked 2022-09-02
If a rock is thrown upward o the planet Mars with a velocity of 10m/s, its height in meters t seconds later is given by y = 10 t 1.86 t 2
a) Find the average velocity over the given time intervals:
i)[1,2] ii)[1,1.5] iii)[1,1.1] iv)[1,1.01] v) [1,1.001]
b) Estimate the instantaneous velocity when t=1.
asked 2022-09-21
Fourth solution of y = 10 3 x y 2 / 5 , y ( 0 ) = 0?
By inspection, we know that y = 0 is a solution. If we separate variables, we get another solution, y = x 10 / 3 . The book tells me this much, and asks me to find the other 2 solutions that differ on every open interval containing x = 0 and are defined on ( , ).
I found the third solution (by guessing) to be y = x 10 / 3 . The only other possibility i could think about is x = 0, but I'm pretty sure that's not valid since y′ would be undefined. What is the last solution? How do we know there are only 4 possible solutions (and not more)?
asked 2022-08-11
Graphing using derivatives
Sketch the graph of the following equation. Show steps of finding out critical numbers, intervals of increase and decrease, absolute maximum and minimum values and concavity.
y = x e x 2
I found the first derivative which is y = ( 2 x 2 + 1 ) e x 2 and I know that in order to find min and max the zeroes for y′ must be found, but y′ doesn't have any real zeroes, and I'm confused about how to go on with solving the problem.
If someone could help me out, that would be appreciated. Thank you in advance.
asked 2022-07-21
Sequence defined by u 0 = 1 / 2 and the recurrence relation u n + 1 = 1 u n 2 .
I want to study the sequence defined by u 0 = 1 / 2 and the recurrence relation u n + 1 = 1 u n 2 n 0..
I calculated sufficient terms to understand that this sequence does not converge because its odd and even subsequences converge to different limits.
In particular we have lim n a 2 n + 1 = 0  and lim n a 2 n + 2 = 1..
Now I can also prove that u n 1 because the subsequent terms are related in this way
f : x 1 x 2 .
In order to prove that the subsequences converge I need to prove either that they are bounded (in order to use Bolzano-Weierstrass) or that they are bounded and monotonic for n > N. Proved that the subsequences converge to two different limits, I will be able to prove that our u n diverges.
Can you help me? Have you any other idea to study this sequence?
asked 2022-09-23
For any n 2 + 1 closed intervals of R , prove that n + 1 of the intervals share a point or n + 1 of the intervals are disjoint
Stuck on a question from 'Introduction to Combinatorics by Martin J. Erickson'.
Q: For any n 2 + 1 closed intervals of R , prove that n + 1 of the intervals share a point or n + 1 of the intervals are disjoint.
I think we can use the Erdos-Szekeres Theorem: relating the usual series of integers to the closed intervals, and the decreasing/increasing monotonic sequences somehow to the two outcomes, but I am stuck on how to technically do this.
Could we measure the 'distance' between the intervals? Creating a decreasing sequence ensuring they'd be close enough to share a point, and an increasing sequence that would ensure they are far enough away to be disjoint?
asked 2022-08-19
Constructing a function that is continuous and has a max on an open interval, but is not necessarily increasing immediately to the left of the max.
Suppose F(x) is continuous on some open interval I and c is a maximum point inside this interval. Is it true that f(x) must be increasing immediately to the left of c and decreasing immediately to the right of c? Proof or counterexample. (Note: A constant function is considered to be both increasing and decreasing.)

New questions