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A non-constant real-valued continuous function (f:R->R) cannot have an arbitirarly small period!

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- Thread starter vidmar
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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.

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A non-constant real-valued continuous function (f:R->R) cannot have an arbitirarly small period!

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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.

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In other words, there is **no** counter-example!

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vidmar said:

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.

Last edited:

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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.

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.

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.

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

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

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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