Prove $x (\sin (x^2))^2$ is continuous but not uniformly continuous.
My proof is below. Can you please verify, critique, or improve?
Continuous. $g(x) = \sin x$ and $g(x) = x^2$ are both continuous, and, given continuous $g$ and $h$, both $g \cdot h$ and $g \circ h$ are continuous.
Not Uniformly Continous. $f$'s amplitude grows without bound, while its period vanishes, so it cannot be uniformly continuous. More formally, $f(x) = 0$ for all $x = \sqrt{\pi n}$, and $f(x) = x$ for all $x = \sqrt{\pi n + \pi/2}$. For any $\delta > 0$, there exists $m$ such that $n > m \implies |\sqrt{\pi n + \pi/2} - \sqrt{\pi n}| < \delta$. Similarly, to state the obvious, for any $\epsilon$, there exists $r$ such that $x > r \implies x > \epsilon$. Thus, for any $\delta > 0, \epsilon > 0$, there exists $x_0, x_1$, such that $|x_1 - x_0| < \delta, f(x_0) = 0, f(x_1) = x_1 > \epsilon$. QED.
The proof is fine, although here are some small critiques.
First, the sentence "Similarly..." does not seem to play any role in the proof. Also, the preceding sentence introduces an irrelevant variable $m$. So I would replace those two sentences by "For any $\delta > 0$ there exists $n$ such that $|\sqrt{\pi n + n/2} - \sqrt{\pi n}| < \delta$" (you should be prepared to defend that statement by proving the existence of $n$, depending on your intended readership). Then immediately follow it with your last sentence.
You could also be more explicit with your last sentence by writing out how to specify $n$, $x_0$ and $x_1$, something like this: "Choose $x_0 = \sqrt{\pi n}$ where $n > $ [whatever quantity is needed to guarantee that $x_1 = \sqrt{\pi n + \pi/2} > \epsilon$, so I guess $n > \left(\epsilon^2 - \pi/2 \right)/\pi$]".
Also, while I do like proofs that start with an intuitive explanation before diving into the logical details, your opening intuitive sentence didn't make any sense to me: What is the amplitude of $f$? What does it mean to say that the period of $f$ vanishes? I was puzzled by that sentence, but eventually ignored it and went on to read the rest of the proof. Afterwards I could go back and see what you kinda/sorta meant in that opening sentence.