bijection from $\mathbb{Q} - \{a\}$ to $\mathbb{Q}$ using elementary functions only?

Question

I was wondering, can you define a bijection from $\mathbb{Q} - \{a\}$ to $\mathbb{Q}$ using elementary functions only ($a \in \mathbb{Q}$)?

Of course there are many set theoretic bijections like that, but I'm looking for a constructive one, a formula, using elementary functions, a smile and a hug if you can do it using field operations only.

Edit: - Elementary functions are field operations, exponentials, logarithms, constants, roots and such. - You may give piecewise functions, but only a finite number of convex pieces.

Answer 1

If $\alpha \in \mathbb{N}$ it's easy! In any other case Consider the following $f:\mathbb{Q}-\{\alpha\}\rightarrow \mathbb{Q}$

$ f(x) = \left\{ \begin{array}{ll} x & \mbox{if $x \notin\mathbb{N}$}\\ \alpha & \mbox{if $x = 0$}\\ x-1 & x\in\mathbb{N}^* \end{array} \right. $