- #31
- 4,662
- 372
ok, what about the question with which iv'e opened the thread?
A question on sequence of functions.
Click For Summary
The discussion revolves around the existence of a continuous, non-periodic function g: R -> R such that for any sequence of real numbers {a_k}, the functions g_k(x) = g(x + a_k) have a uniformly converging subsequence on R. Participants explore examples like g(x) = sin(e^x) and g(x) = exp(-x^2), debating their properties and the implications of the Arzelà-Ascoli theorem. The consensus suggests that while finding such a function is challenging, examples like sin(x) + cos(πx) and 'almost periodic' functions may satisfy the criteria. The conversation highlights the nuances of periodicity and equicontinuity in the context of uniform convergence. Ultimately, the group concludes that while difficult, the existence of such a function is plausible.