Is the function $ f \colon X \to f(X), f(x) =x^3 $ continuous, uniformly continuous

Question

Let $$ X = \big( [0,1] \cap \mathbb{Q} \big) \cup \left\{ 1+ \frac1n \middle|\ n \in \mathbb{N} \right\} $$ be a subspace of $ \mathbb{R} $. Is the function $$ f \colon X \to f(X), f(x) =x^3 $$ continuous, uniformly continuous? It kinda seems like it is continuous because it is an elementary function. I'm stuck.

Answer 1

Function $x^3$ is uniformly continuous on any compact subset of $\mathbb{R}$ and hence on any bounded subset.

Answer 2

If $x,y \in X$ then $0 \le x,y \le 2$ and therefore

$$|f(x)-f(y)|=|x-y| (x^2+xy+y^2) \le |x-y|(4+4+4 )= 12|x-y|.$$