Monday, 11 February 2019

calculus - Find a Continuous Function











I am having a problem with this exercise. Could someone help?



Suppose a(0,1) is a real number which is not of the form 1n for any natural number n
n. Find a function f which is continuous on [0,1] and such that f(0)=f(1) but which does not satisfy f(x)=f(x+a) for any x with x, x+a[0,1].




I noticed that this condition is satisfied if and only if f(x)f(0)



Thank you in advance


Answer



Look at f(x)=sin(2πx). For which values of a can you find an x[0,1] with x+a[0,1] and f(x)=f(x+a)? In particular, if you additionally require a>12, can such an a exist at all?



Once you've answered that you've solved your problem for some values of a. Which are those?



A general solution can be found in the answers to Universal Chord Theorem (Link found by the user who asked the question). To quote f(x)=sin2(πxa)x sin2(πa)



is a solution. This works because f(x)=f(x+a) implies asin2(πa)=0 and thus a=1n for some nN. The answers to the linked questions also prove that a1n for every nN is a necessary condition for a solution to exist.


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