Let f be a continuous function defined on the interval [0,1] such that f(0) = f(1). show that there exists a number 0 < = a < = 1/2 such that f(a) = f(a+1/2)

Explanation:

In order to prove that affirmation, we define the function g over the interval [0, 1/2] with the formula

If we evaluate g at the  endpoints we have

g(0) = f(1/2)-f(0) = f(1/2) - f(1) (because f(0) = f(1))  

g(1/2) = f(1) - f(1/2) = -g(0)

Since g(1/2) = -g(0), we have one chance out of three

We will prove that g has a zero on [0,1/2]. If g(0) = 0, then it is trivial. If g(0) ≠ 0, then we are in one of the first two cases, and therefore g(0) * g(1/2) < 0. Since f is continuous, so is g. Bolzano's Theorem assures that there exists c in (0,1/2) such that g(c) = 0.  This proves that g has at least one zero on [0,1/2].

Let c be a 0 of g, then we have

Hence, f(c+1/2) = f(c) as we wanted.

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