does the uniform continuity of $f$ implies uniform continuity of $f^2$ on $\mathbb{R}$?
my question is if $f:\mathbb{R}\rightarrow\mathbb{R}$ is uniformly continuous, does it implies that $f^2$ is so?and in general even or odd power of that function?
No. For example, $f(x)=x$ is uniformly continuous, but $f(x)=x^2$ is not, and neither for that matter is $f(x)=x^n$ for any $n>1$.
No. If $f(x)= x$ then $f^2(x)$ is not unifomly continuous. A statement like that will only hold on compact sets.