Interesting problem from my analysis class

In summary, the conversation discusses the proof of the existence of a horizontal chord of length 1/n in the graph of a continuous function f on [0,1] with f(0) = f(1). The hint is to write f(1) - f(0) as a telescoping sum. It is mentioned that this problem may be considered trivial for all numbers less than or equal to 1, not just 1/n, and that it follows from the intermediate value theorem. The details of the proof are considered tedious but the problem itself is considered cute. The assumption is made that f(x) is non-negative.
  • #1
StonedPanda
60
0
Let n be a positive integer and suppose [tex]f[/tex] is continuous on [tex][0,1][/tex] and [tex]f(0) = f(1)[/tex]. Prove that the graph of [tex]f[/tex] has a horizontal chord of length [tex]1/n[/tex]. In other words, prove there exists [tex]x \in [0,(n - 1)/n][/tex] such that [tex]f(x+1/n) = f(x)[/tex]
 
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  • #2
no one even wants to try?
 
  • #3
hint: write f(1) - f(0) as a telescoping sum.
 
  • #4
StonedPanda said:
no one even wants to try?

I would hazard a guess that people are suspicious that this is your homework.
 
  • #6
seems trivial. indeed trivial for all numbers less than or equal to 1, not just 1/n.

after thinking about an actual proof for a few minutes let me rephrase that as "intuitively plausible", rather than "trivial".

it apparently follows from the intermediate value theorem but the details seem tedious, even elusive. cute problem.
 
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  • #7
assume f(x) non negative, then what?
 

What is an interesting problem from analysis class?

An interesting problem from analysis class is a mathematical or scientific question that requires critical thinking and problem-solving skills to solve. It may involve using various analytical techniques, such as calculus, statistics, or data analysis, to find a solution.

Why is it important to study interesting problems in analysis class?

Studying interesting problems in analysis class helps develop critical thinking, problem-solving, and analytical skills that are essential for scientific research and other fields. It also allows students to apply theoretical knowledge to real-world problems, which can be beneficial in their future careers.

What are some examples of interesting problems from analysis class?

Examples of interesting problems from analysis class may include finding the maximum and minimum values of a function, determining the area under a curve, analyzing patterns in data sets, or solving optimization problems.

How do you approach solving an interesting problem from analysis class?

To solve an interesting problem from analysis class, it is essential to first understand the problem and identify the key variables and concepts involved. Then, you can apply relevant analytical techniques and equations to find a solution. It is also important to double-check your work and make sure your solution is reasonable and accurate.

What skills can be gained from solving interesting problems in analysis class?

Solving interesting problems in analysis class can help develop critical thinking, problem-solving, logical reasoning, and mathematical skills. It also helps improve data analysis and interpretation skills, which are valuable in various fields, such as science, engineering, finance, and business.

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