Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function $f$, where $f$ satisfies a certain number of assumptions to be determined.
I have already thought about $f$ being $C^1$ and bounded for example, but this doesn't seem to work as I can still find some counter-examples based on trigonometric functions.
I guess my question would therefore be: what are the least constraining assumptions having to be made on $f$ (and maybe even on $u$) which would ensure uniform convergence of $f_n$ to $f$ on $\mathbb{R}_+$? Does the assumption of bounded first derivative suffice (as per comment below)?