A chord at the edges of a graph

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    Chord Graph
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Discussion Overview

The discussion revolves around a mathematical exercise concerning the existence of chords in the graph of a continuous function defined on the interval [0,1], specifically focusing on proving the existence of chords of certain lengths. The scope includes theoretical exploration and mathematical reasoning.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants clarify the definition of a chord as a line segment joining two points on the graph of a function, questioning the terminology used in the original post.
  • One participant suggests that for any continuous function on the interval [a, b], chords exist of all lengths up to a certain maximum, proposing a general result that could be derived from the definition of continuity.
  • Another participant mentions a specific method involving the function F_n(x) and the Bolzano theorem to prove the existence of chords of length 1/n, indicating a different approach from what was previously discussed.
  • There is a request for clarification on the method proposed by another participant, highlighting a lack of understanding regarding the derivation of a specific inequality related to chord lengths.
  • Some participants express confusion about the terminology and mathematical concepts, indicating a need for clearer explanations.

Areas of Agreement / Disagreement

Participants express differing views on the terminology and methods used to approach the problem. There is no consensus on a single method or understanding of the definitions involved, and the discussion remains unresolved regarding the best approach to proving the existence of chords of specified lengths.

Contextual Notes

There are limitations in the clarity of definitions and terminology, particularly regarding the term "chord" and the notation "C_f," which is specific to a certain cultural context. The discussion also reflects varying levels of understanding of the mathematical concepts involved.

Kostas Tzim
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Greetings, i found an interesting exercise from my perspective, it's not about HW, i just want to see different approaches than the Greek math forum i posted yesterday, so we have:

If $$ f $$ is a function, then a chord is a straight portion whose edges belong to $$ C_f $$
f is a continuous function. its domain is $$ [0,1] $$ and $$ f(0)=f(1)=0 $$

A) Prove that a chord with length $$ \tfrac{1}{2} $$ exists
B) Prove that a chord with length $$ \tfrac{1}{n} $$ exists where n=1,2,3..

ps: (sorry for the ugly latex appearance), i also think that A) question is a result of B)
 
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Kostas Tzim said:
Greetings, i found an interesting exercise from my perspective, it's not about HW, i just want to see different approaches than the Greek math forum i posted yesterday, so we have:

If f is a function, then a chord is a straight portion whose edges belong to [/itex]C_f[/itex]
f is a continuous function. its domain is [0,1] and f(0)=f(1)=0

A) Prove that a chord with length \tfrac{1}{2} exists
B) Prove that a chord with length tfrac{1}{n} exists where n=1,2,3..

ps: (sorry for the ugly latex appearance), i also think that A) question is a result of B)
On this board, use [ itex ] and [ /itex ] (without the spaces) rather than the dollar signs to get "in line" Latex.
Does "C_f" mean the graph of f? I don't believe that is standard notation. Let g(x) be the length of the chord from (0, 0) to (x, f(x)). Show that g is a continuous function. Therefore g takes on all values between 0 (when x= 0) and 1 (when x= 1).
 
I think the translation is confusing: a chord is normally defined as a line segment joining two points on the graph of a function. The words "portion" and "boundary" are not useful here.

Assuming this is what you mean, the more general result that for any function f continuous in the interval [a, b] chords exist of all lengths ## l; 0 < l \le \sqrt {(f(b)-f(a))^2 + (b-a)^2} ## can be proved from the definition of a continuous function. Can you see how?
 
Thanks for your answer..Yes the term "portion" is bad sorry :/ . The solution i got from the other forum was different : Assume the function F_n(x)=f(x+\dfrac{1}{n})-f(x) and then i use the Bolzano theorem if Fn(a)F_n(b)&lt;0, (sorry i find it extremely hard to translate some greek terms we use) for the second case we follow an inequallity method

Could you explain me your method? i can't see very clearly how you ended up with this specific inequallity
 
Last edited:
#https://www.physicsforums.com/members/hallsofivy.331/ you are right the C_f is a symbol we use in greece, it means the graph of f
 
Last edited:
Kostas Tzim said:
Could you explain me your method? i can't see very clearly how you ended up with this specific inequallity
How long is the chord joining ## f(a) ## and ## f(b) ##? Note that you can also use two # (hash) characters as a short cut to bracket inline ## \LaTeX ## on this forum.
 
I didnt understand this..what do you mean by how long? sometimes my brains completely stops working :P
 
Kostas Tzim said:
I didnt understand this..what do you mean by how long? sometimes my brains completely stops working :P
"How long" here means "what is the length of". The chord is the hypotenuse of a right angled triangle.
 

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