# A chord at the edges of a graph

1. Aug 23, 2015

### 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)

2. Aug 23, 2015

### HallsofIvy

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).

3. Aug 23, 2015

### MrAnchovy

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?

4. Aug 23, 2015

### 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 cant see very clearly how you ended up with this specific inequallity

Last edited: Aug 23, 2015
5. Aug 23, 2015

### Kostas Tzim

#HallsofIvy you are right the $C_f$ is a symbol we use in greece, it means the graph of $f$

Last edited: Aug 23, 2015
6. Aug 23, 2015

### MrAnchovy

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.

7. Aug 23, 2015

### Kostas Tzim

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

8. Aug 23, 2015

### MrAnchovy

"How long" here means "what is the length of". The chord is the hypotenuse of a right angled triangle.