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A chord at the edges of a graph

  1. Aug 23, 2015 #1
    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)
     
  2. jcsd
  3. Aug 23, 2015 #2

    HallsofIvy

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    On this board, use [ itex ] and [ /itex ] (without the spaces) rather than the dollar signs to get "in line" Latex.
    Does "[itex]C_f[/itex]" 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).
     
  4. Aug 23, 2015 #3
    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?
     
  5. Aug 23, 2015 #4
    Thanks for your answer..Yes the term "portion" is bad sorry :/ . The solution i got from the other forum was different : Assume the function [itex] F_n(x)=f(x+\dfrac{1}{n})-f(x) [/itex] and then i use the Bolzano theorem if [itex] Fn(a)F_n(b)<0 [/itex], (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 cant see very clearly how you ended up with this specific inequallity
     
    Last edited: Aug 23, 2015
  6. Aug 23, 2015 #5
    #HallsofIvy you are right the [itex]C_f[/itex] is a symbol we use in greece, it means the graph of [itex]f[/itex]
     
    Last edited: Aug 23, 2015
  7. Aug 23, 2015 #6
    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.
     
  8. Aug 23, 2015 #7
    I didnt understand this..what do you mean by how long? sometimes my brains completely stops working :P
     
  9. Aug 23, 2015 #8
    "How long" here means "what is the length of". The chord is the hypotenuse of a right angled triangle.
     
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