A chord at the edges of a graph

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SUMMARY

The discussion centers on proving the existence of chords of specific lengths in the context of continuous functions defined on the interval [0,1] with endpoints at zero. Participants explore two main problems: A) demonstrating that a chord of length 1/2 exists, and B) proving that chords of length 1/n exist for n=1,2,3. The concept of a chord is clarified as a line segment connecting two points on the graph of the function, denoted as C_f. The Bolzano theorem is referenced as a method for establishing the existence of these chords.

PREREQUISITES
  • Understanding of continuous functions and their properties.
  • Familiarity with the concept of chords in the context of graph theory.
  • Knowledge of the Bolzano theorem and its application in proving the existence of roots.
  • Basic proficiency in LaTeX for mathematical notation.
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  • Study the Bolzano theorem and its implications for continuous functions.
  • Learn about the properties of chords in graph theory and their mathematical definitions.
  • Explore the concept of continuity and its role in determining the existence of specific lengths of chords.
  • Practice using LaTeX for mathematical expressions to enhance clarity in discussions.
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Mathematicians, students studying calculus or real analysis, and anyone interested in the properties of continuous functions and their graphical representations.

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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