Continous function in interval problem

In summary, the conversation discusses the proof of the existence of x and y with the given property in a continuous function f on the interval [0,2]. The Bolzano-Cauchy theorem is used to prove this, with careful consideration of special cases. The presentation of the proof is thorough and well thought out, with a minor suggestion to use a smaller symbol instead of QUOD ERAT DEMONSTRANDUM.
  • #1
nuuskur
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Homework Statement


Let [itex]f: [0,2]\to\mathbb{R}[/itex] be continuous and [itex]f(0) = f(2)[/itex]. Show that there exist [itex]x,y\in [0,2][/itex] with the following property:
[itex](*)\ y-x = 1[/itex] and [itex]f(x) = f(y)\ (*)[/itex]

Homework Equations


Bolzano-Cauchy theorem: If a function [itex]f[/itex] is continuous in some interval [itex][a,b][/itex] and [itex]f(a) <0, f(b) > 0[/itex] (or vice versa) then there exists [itex]c\in (a,b)[/itex] such that [itex]f(c) = 0[/itex]

The Attempt at a Solution


If [itex]f[/itex] was constant, then it's trivial. Fix [itex]x\in [0,1][/itex] and the condition [itex]f(x+1)-f(x) = 0[/itex] is satisfied.
Hence, assume [itex]f[/itex] is not constant.

Let us observe function [itex]g(x) := f(x+1)-f(x)[/itex], [itex]0\leq x\leq 1[/itex]. The objective is to show that [itex]g(x)=0[/itex] is possible with which we will have proven the existence of the required [itex]x,y[/itex] (is this correct to say? )

Let us note that:
[itex]g(0) = f(1) - f(0)[/itex] and [itex]g(1) = f(2) - f(1) = f(0) - f(1)[/itex]. If [itex]g(0) > 0[/itex], then [itex]g(1) <0[/itex] (or vice versa), we can therefore conclude that:
Per Bolzano-Cauchy theorem there exists [itex]c\in (0,1)[/itex] such that [itex]g(c) = f(c+1) - f(c) = 0[/itex] from which we can establish [itex]x = c[/itex] and [itex]y = c+1[/itex] and the condition [itex](*)[/itex] is satisfied

If [itex]g(0) = 0[/itex] then also [itex]g(1) = 0[/itex] and again the condition [itex](*)[/itex] is satisfied. [itex]Q.E.D[/itex]
 
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  • #2
Looks nice. And what exactly do you expect as replies here? Does your proof still hold for arbitrary c ∈[0,d] with y - x = c and arbitrary continuous functions f : [a,b] → ℝ with f(a) = f(b) and how big can d be at most?
 
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  • #3
Most of all expect criticism on presenting proof, generalizing the problem is welcome. Also would like the proof to be challenged if there is something I might have missed.
 
  • #4
nuuskur said:
Most of all expect criticism on presenting proof, generalizing the problem is welcome. Also would like the proof to be challenged if there is something I might have missed.
I've seen nothing wrong. And your presentation reveals that you work carefully and think about the special cases. I'ld let it go through. (Bluster me if I'm wrong!) There is only one little, tiny, small remark from my side: Don't shout QUOD ERAT DEMONSTRANDUM. A simpe qed will do and a ◊ or box is even more pleasant
 
  • #5
nuuskur said:
Most of all expect criticism on presenting proof, generalizing the problem is welcome. Also would like the proof to be challenged if there is something I might have missed.
I've seen nothing wrong. And your presentation reveals that you work carefully and think about the special cases. I'ld let it go through. (Bluster me if I'm wrong!) There is only one little, tiny, small remark from my side: Don't shout QUOD ERAT DEMONSTRANDUM. A simpe qed will do and a ◊ or box is even more pleasant. :wink:

- sorry, probs with the connection and a failed search for a remove button
 
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What is a continuous function?

A continuous function is a mathematical function that does not have any sudden changes or breaks in its graph. This means that the function can be drawn without lifting the pen from the paper.

What is an interval in mathematics?

In mathematics, an interval is a set of real numbers between two given values. It can be represented by using brackets or parentheses, such as [a, b] or (a, b). The endpoints of an interval can be included or excluded, depending on the notation used.

How do you determine if a function is continuous on a given interval?

A function is considered continuous on a given interval if it is defined at every point within that interval and if the limit of the function at the endpoints of the interval is equal to the value of the function at those endpoints. This is known as the continuity theorem.

What is the importance of continuous functions in mathematics?

Continuous functions are important in mathematics because they allow us to model and analyze real-world phenomena. They are also used in many areas of mathematics, such as calculus, where they are used to calculate derivatives and integrals.

Can a function be continuous at some points and discontinuous at others?

Yes, a function can be continuous at some points and discontinuous at others. This type of function is known as a piecewise continuous function. It is defined differently on different intervals, and there may be a break or jump in the graph at the endpoints of those intervals.

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