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I am not getting satisficatory explanation for this. Clearly $f(x+T) = f(x)$ for all values of $T$.

If we assume it is periodic, does this mean period = $0$?

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What it means is that any number is a period. There is no "minimum" period though.

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    $\begingroup$ thank you ! that clears up the confusion :) so this looks like a degenerate case of periodic functions where the function satisfies the definition but the graph doesn't look periodic(example : y^2=x^2 is a conic by definition but the graph is just a pair of intersecting lines) $\endgroup$ – AgentS Aug 23 '14 at 17:03
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    $\begingroup$ $$\text{Harry Potter shouts: The minimum period is $h$ tends to zero!}\\ \text{Dumbledore retorts: ... or }\frac{1}{\infty} $$ $\endgroup$ – Nick Aug 23 '14 at 17:10
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Yes, every constant function is periodic, and when you look at the definition of a periodic function with period T (see here) then it's easy to see that a constant is periodic with any positive number as period. So $f(x)=10$ is $n-$ periodic for every $n\in \mathbb{N}$

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    $\begingroup$ No need to restrict $n$ to $\Bbb N$; it is periodic for every $n\in \Bbb R$, in fact. $\endgroup$ – MJD Aug 23 '14 at 17:02
  • $\begingroup$ Yes, you are right. $\endgroup$ – Barkas Aug 23 '14 at 17:04
  • $\begingroup$ thanks! that wiki link says period has to be a positive constant. however allowing negative/complex periods should be okay i guess concept wise - its just a matter of how we define after all ;) $\endgroup$ – AgentS Aug 23 '14 at 17:13

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