Collection of functions $f: \mathbb{R} \to \mathbb{R}$ is uncountable.

Question

Let $F$ be the collection of all functions $f: \mathbb{R} \to \mathbb{R}$. Prove that $F$ is uncountable.

Proof: Let $r \in \mathbb{R} $. Then the function $f: x \to x^r$ is in $F$. Since the reals themselves are uncountable, we have uncountably many functions of the type $x^r$.

I'm unsure wether my argument is viable. What do you think?