Q:

Is $f^2$ is uniformly continuous

Let $f$ be uniformly continuous in $I$. Prove or disprove by an example that $f^2$ is uniformly continuous in $I$.

A:

According to the well stated comment of @Nilknarf, if $f^2=(f)^2$ then for $I=[0, + \infty)$ and $f(x)=x$ we have the counterxample.
For a proof of this counterexample  we can use the sequential property of uniform continuity by taking the sequences: $$x_n=n + \frac{1}{n}$$ $$y_n=n$$
If the O.P wants the case where $f^2=fof$ then we know that the composition of uniformly continuous functions is uniformly continuous.

