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Given the following definition of a periodic function:

$$\exists P, P > 0, f(x + P) = f(x)$$

It is possible to argue that $f(x)=k$ ($k$ being a constant) is a periodic function, since you can define $P$ to be any given constant within the real numbers and the definition would be valid. ($f(x + P) = f(x)$ will always be true if $P\in \mathbb R$).

So my question is, can a horizontal line be considered periodic, even when its period is ambiguous?

Jonas Meyer
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Juan Camilo Osorio
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  • Periods are ambiguous in other cases. For example, $\sin \theta$ has periods $2\pi k$ for any $k \in \mathbb{Z}$. Of course, you could pick a minimal period in that case. But, I don't see the problem. I mean, the constant function is part of the family of periodic functions built over any period. This is a good thing. – James S. Cook Sep 15 '14 at 02:22
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    Based on the definition of periodic, it looks that way! – layman Sep 15 '14 at 02:23
  • Yes I would say so under this definition. If you want something less trivial, it might behoove one to ask that there exists a smallest $P>0$ such that $f(x+P)=f(x)$ for all $x$. – Cameron L. Williams Sep 15 '14 at 02:23

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