# Is $f(x)=10$ a periodic function?

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 = $0$?

• It just means that it is periodic for any value of $T$. – Daniel Aug 23 '14 at 16:57
• en.wikipedia.org/wiki/Periodic_function – Surb Aug 23 '14 at 16:57
• We do not normally consider $0$ to be a possible period of a periodic function; if we did, then every function would be periodic. – MJD Nov 1 '14 at 20:52
• – Przemysław Scherwentke Nov 8 '14 at 3:17

## 2 Answers

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

• 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) – AgentS Aug 23 '14 at 17:03
• $$\text{Harry Potter shouts: The minimum period is h tends to zero!}\\ \text{Dumbledore retorts: ... or }\frac{1}{\infty}$$ – Nick Aug 23 '14 at 17:10

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 $n\in \mathbb{N}$

• No need to restrict $n$ to $\Bbb N$; it is periodic for every $n\in \Bbb R$, in fact. – MJD Aug 23 '14 at 17:02
• Yes, you are right. – Barkas Aug 23 '14 at 17:04
• 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 ;) – AgentS Aug 23 '14 at 17:13