Periods of continuous functions

In summary, the OP asked for a proof that a non-constant real-valued continuous function cannot have an arbitirarly small period, but Antiphon provides a counter-example.
  • #1
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I would like a proof or a counter-example for the following claim:
A non-constant real-valued continuous function (f:R->R) cannot have an arbitirarly small period!
 
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  • #2
Proof: Suppose we have a non-constant function f:R->R which has arbitrary small period, that is: for each [tex]\delta > 0[/tex] f is periodic with some period [tex]0 < p < \delta[/tex].

Let [tex]x \in \mathcal{R}[/tex]. Because f is non-constant there is a [tex]y \in \mathcal{R}[/tex] such that [tex]f(x) \neq f(y)[/tex]. Now we claim that f takes the value f(y) on every neighbourhood [tex]\left]x-\delta,x+\delta\right[[/tex] of x with [tex]\delta > 0[/tex]. Proof: f is periodic with some period [tex]0 < p < \delta[/tex]. There exists an integer n such that [tex]y-np \in \left]x-\delta,x+\delta\right[[/tex] and [tex]f(y-np)=f(y)[/tex].

Hence f cannot be continuous at x.
 
  • #3
In other words, there is no counter-example!
 
  • #4
vidmar said:
I would like a proof or a counter-example for the following claim:
A non-constant real-valued continuous function (f:R->R) cannot have an arbitirarly small period!

Counter-example: [tex] f(x) = \lim_{h\rightarrow{0}}sin(x/h) [/tex].

This works for any finite h. But I don't think the limit exists, so this
is consistent with the proof given by Timbuqtu.
 
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  • #5
Antiphon said:
Counter-example: [tex] f(x) = \lim_{h\rightarrow{0}}sin(x/h) [/tex].

This works for any finite h. But I don't think the limit exists, so this
is consistent with the proof given by Timbuqtu.

An incredibly important distinction must be made: between

*for arbitrarily small period p, --> there exists a function which is periodic with period P, 0<P<p

and

**there exists a function f --> for which, for arbitrarily small p, f has a period P, 0<P<p

The two statements are independent. * is true. ** is false. The OP asked about **, which is false (per Timbuqtu's proof). Of course, Antiphon successfully proves that * is true.

o:)
 

What are periods of continuous functions?

Periods of continuous functions refer to a specific interval of a function's input or domain that repeats itself, resulting in a repetitive pattern in the output or range.

How can we identify periods of continuous functions?

To identify periods of continuous functions, we can look for a specific value of the input that results in the same output. This value is known as the period and can be found by analyzing the function's equation or graph.

Are all continuous functions periodic?

No, not all continuous functions are periodic. Some continuous functions have a constantly changing output and do not exhibit a repeating pattern.

Can a function have multiple periods?

Yes, a function can have multiple periods. In fact, some functions have infinite periods, meaning that the function repeats itself indefinitely.

How are periods of continuous functions useful in real-life applications?

Periods of continuous functions are useful in many real-life applications, such as in modeling and predicting natural phenomena like weather patterns or economic cycles. They are also used in signal processing and communication systems to analyze and filter repetitive signals.

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