A chord at the edges of a graph

[Kostas Tzim]

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)

 

[HallsofIvy]

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).

 

[pbuk]

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?

 

[Kostas Tzim]

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 $Fn(a)F_n(b)<0$, (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

 

[Kostas Tzim]

#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 $f$

 

[pbuk]

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.

 

[Kostas Tzim]

I didnt understand this..what do you mean by how long? sometimes my brains completely stops working :P

 

[pbuk]

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.