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I have a problem: g(x) is an even periodic function with ${T} > {0}$. Show that there exists ${c}$ with ${0<c<T}$ such that $${g}(c+x)={g}(c−x)$$for every real number {x}.

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1 Answer 1

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If $c=\frac{T}{2}$ then

$$ g(c+x)=g(c+x-2c)=g(x-c)\overset{\text{ g is even}}{=}g(c-x) $$

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