real analysis - Is there a bijection from [0,1] to R?

I'm looking for a bijection from the closed interval [0,1] to the real line. I have already thought of $\tan(x-\frac{\pi}{2})$ and $-\cot\pi x$, but these functions aren't defined on 0 and 1.

Does anyone know how to find such a function and/or if it even exists?

Thanks in advance!

Answer

Use Did's method here to construct a bijection $[0,1] \to (0,1)$. Play around with $\tan$ for a bijection $(0,1) \to \mathbb{R}$

Note that any bijection cannot be continuous. This is because $[0,1]$ is compact.