The range of any continuous function from $[0, \omega_1] \times [0, \omega_1] \rightarrow \mathbb{R}$ is countable

Question

I came across this question: “Prove that the range of any continuous function from $[0, \omega_1] \times [0, \omega_1] \rightarrow \mathbb{R}$ is countable” I’m quite new to ordinals and I’m having a lot of trouble wrapping my head around this statement, much less the proof of it. Here, $[0,\omega_1]$ is the second uncountable, so im very unclear on how its range could possible be countable? In this problem $\mathbb{R}$ has the standard topology and the product has the product topology. Any intuition or help would be greatly appreciated!