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Let $f:ℂ→ℝ$ be a non-analytic function. Let $h(s)=f(a+s)$, where $a∈ℝ$. If $h$ is an odd function, then $(0,0)$ is a symmetric point for $h$ (the graph of $f$ is symmetric with respect to the origin). Is it true that the point $(a,0)$ is a symmetric point for $f$?

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Sure. The function $h$ is $f$ translated $a$ units to the left, so the point $(a,0)$ on $f$ corresponds to $(0,0)$ on $h$. –  01000100 Jan 1 '13 at 9:24
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