A question on sequence of functions.

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Homework Help Overview

The discussion revolves around the existence of a continuous function g: R -> R that is not periodic, yet for every sequence of real numbers {a_k}, the sequence of functions g_k(x) = g(x + a_k) has a subsequence that converges uniformly on R. Participants explore various examples and properties of functions, particularly focusing on continuity, boundedness, and equicontinuity.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Some participants propose specific functions, such as g(x) = sin(e^x) and g(x) = exp(-x^2), while questioning their suitability regarding periodicity and uniform convergence. Others discuss the implications of boundedness and equicontinuity in relation to the Arzelà-Ascoli theorem. There are inquiries about the nature of periodic functions and the definitions surrounding them.

Discussion Status

The discussion is ongoing, with participants examining different functions and their properties. Some have expressed uncertainty about the correctness of their examples, while others are exploring the implications of boundedness and continuity. There is no explicit consensus on the existence of such a function, and various interpretations are being considered.

Contextual Notes

Participants are grappling with definitions of periodicity and the conditions under which functions can be considered equicontinuous. There are also references to the limitations of applying the Arzelà-Ascoli theorem in non-compact spaces like R.

  • #31
ok, what about the question with which iv'e opened the thread?
 
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  • #32
loop quantum gravity said:
i thought a periodic function has a small period which is non zero, here clearly there isn't such period.
anyway, if this is considered a periodic function then back to the original question is there such a function which is not periodic?

I think NateTG's example sin(x)+cos(pi*x) is an excellent example of a function that statisfies your requirements (as are 'almost periodic' functions in general). How many times has that been stated on this thread? Can you prove it?
 
Last edited:
  • #33
i think by arzela ascoli theorem i only need to show that it's uniformly bounded (obviously it is by: 2. and that this sequence of functions is equicontinuous, which is also bacause of the first derivative is uniformly bounded so it's equicontinuous.

ok, thanks Dick,Nate and also Matt, i appreciate your help on this question.
 

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