if $|f(k) | le k$ for all integers $k$, does that mean $ |f(x)| le |x|$ for all $x$ in $mathbb R$?

if $|f(k) | le k$ for all integers $k$, does that mean $ |f(x)| le |x|$ for all $x$ in $mathbb R$?

Note that $f$ is uniformly continuous.

This question is a follow up to a previous answer: http://math.stackexchange.com/a/1716695/326454 so this answer is wrong write?