Give an example of a function $f: X \to Y$ between metric spaces which is continuous but not uniformly continuous, and explain why it is not uniformly continuous.
My intuition was $f: [0,1] \to [0,1]$ given by $f(x) = x^2$. I think I was able to show that $f$ is continuous, but I was having trouble proving $f$ was not uniformly continuous
If you want to show non-uniform continuity, you will have to have an unbounded domain. This is because $f:[a,b] \to \mathbb{R}$ being continuous implies $f$ is uniformly continuous.
Modifying your example, $f:[0,\infty) \to \mathbb{R}$ where $f(x) = x^2$ is not uniformly continuous.
Suppose that $f(x) = x^2$ is uniformly continuous, then for $\epsilon = 1$, there exists a $\delta >0$ such that $|x-y|<\delta$ implies $|x^2 - y^2|<1$.
Looking at the graph of $f$ we guess that $x = R$ is a large value. We take $y = R + \frac{\delta}{2}$ so that $|x-y| < \delta$.
Then, $$1 >|f(x)-f(y)| = |x^2 - y^2| = |R^2 - (R+\frac{\delta}{2})^2| = |R\delta + \frac{\delta^2}{4}|$$.
But clearly, if we take $R$ big enough this value will be greater than 1. Hence, we have a contradiction and $f$ is not uniformly continuous.