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# [SOLVED] Sum of periodic functions

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 Subject: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 9:18 pm Let m(x) and n(x) be two continuous functions R-->R. Suppose m(x) is periodic with period M and n(x) is periodic with period N. Show that m(x)+n(x) is periodic if and only if M/N is rational.Last edited by Bruno on Wed Nov 11, 2009 5:51 pm; edited 1 time in total

Descartes

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 Subject: Re: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 9:54 pm Sol: simply solvedif M/N be an irrational number then the set {mM-nN where m,n\in \mathbb{N}} is dense in \mathbb{R}, so you then get functions m and n just differs by a constant and more M=N which is of course a contradiction.

Posts : 184
Join date : 2009-09-15
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Location : the infinite, frictionless plane of uniform density

 Subject: Re: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 10:15 pm It's good, I think you have the general idea, but you don't give much details! Where do you use the continuity of m and n?

Descartes

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 Subject: Re: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 10:21 pm suppose that m_kM-n_kN is convergent to x_0 so you get m(x_0)+n(x_0)= lim {m(m_kM-n_kN)+n(m_kM-n_kN)}=lim{m(-n_kN)+n(m_kM)}=m(\infty)+n(\infty)=const.

Descartes

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 Subject: Re: [SOLVED] Sum of periodic functions   Sun Nov 08, 2009 10:27 pm , by the way, your proof today was just awesome