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**Haven** Let $\displaystyle f$ be a continuous function on $\displaystyle \mathbb{R}$. Let $\displaystyle f_n = f(nt)$ for $\displaystyle n \in \mathbb{N}$ be equicontinuous on [0,1]. I.e., $\displaystyle \forall \epsilon >0, \exists \delta >0 $ such that if $\displaystyle |x-y| < \delta$, then $\displaystyle |f_n(x) - f_n(y)| < \epsilon$ for all n.

What can we conclude about $\displaystyle f$?

All I am able to get is that since each $\displaystyle f_n$ is defined on a compact set, then each $\displaystyle f_n$ is pointwise bounded. So, $\displaystyle \{f_n \}$ is uniformly bounded. So, $\displaystyle f$ is uniformly bounded.

Is there something else that I can conclude?