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Does there exist a continuous function g:R->R that is not periodic, and for every sequence of real numbers {a_k} the sequence of functions g_k(x)=g(x+a_k) has a subsequence converging uniformly on R.

i think i found a suitable function, i just need to be sure it's fine:

i defined g_k(x)=g(x)=sin(e^x)

this function isn't periodic, but if i use Arzela-Ascoli lemma in here, then first of all, this function is uniformly bounded by 1, and i can show that it's equicontinuous, i mean all i need to show is that:

for every e'>0, there exist d>0, such that for every x,y in R whihc satisfy

|x-y|<d and for every n then |f_k(x)-f_k(y)|<e'.

here i can use the fact that |sin(e^(a_k+x))-sin(e^(a_k+y))|<=|e^a_k||e^x-e^y|

e^a_k converges, and e^x is continuous, so we can prove the equality from this, am i correct?

thanks in advance.