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Let f(x) be periodic and k be a positive...

Let `f(x)` be periodic and `k` be a positive real number such that `f(x+k)+f(x)=0fora l lx in Rdot` Prove that `f(x)` is periodic with period `2kdot`

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We have `f(x+k)+f(x)=0 AA x in R`
or ` f(x+k)= -f(x) AA x in R`
Put `x=x+k.` Then,
or `f(x+2k)= -f(x+k) AA x in R`
or ` f(x+2k) = f(x), AA x in R " "[ "As "f(x+k)= -f(x)]`
which clearly shows that f(x) is periodic with period 2k.
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