Prove or disprove: There exists a family of continuous functions $f_n:[0,1]\rightarrow \mathbb{R}$, $n\in \mathbb{N}$ with graphs $T(f_n)$ such that $[0,1]\times \mathbb{R}=\cup_{n\in \mathbb{N}}T(f_n)$.
I claim that there is no such a family of functions. By contradiction suppose there exists such a family of functions. So ${0}\times \mathbb{R}=\cup_{n\in \mathbb{N}} (0,f_n(0))$. So $\mathbb{R}$ is a countable unions of real numbers. Contradiction.
However, I didn't use the assumption of continuity in the problem. I'm doubtful my solution is correct.