my question reads as:
Let $\mathcal R[0,1]$ denote the set of all real-valued functions from $[0,1]$ to $\mathbb R$ and let $\mathcal C[0,1]$ denote the set of continuous functions on $[0,1]$.
i) Prove that $|\mathcal C[0,1]|=|\mathbb R$|.
ii) Prove that $|\mathcal R[0,1]|>|\mathbb R$|.
I've been fairly confident in my work with cardinality, but with a set containing functions, I am finding it hard to get a lead.
Thank you in advanced for your help.