Let me give the reason I ask this question. We know that for a point $p = (x^1, \dots, x^n) \in \mathbb{R}^n$, the tangent space at $p$ denoted by $T\_p(\mathbb{R}^n)$ has as basis $$\left\{\frac{\partial}{\partial x^1}\bigg|\_p, \dots, \frac{\partial}{\partial x^n}\bigg|\_p\right\}$$

where $$\frac{\partial}{\partial x^i}\bigg|\_p \text{ is defined by } \left(\frac{\partial}{\partial x^i}\bigg|\_p\right)(f) = \frac{\partial f}{\partial x^1}(p)$$

Now my question is what **exactly** are these: $$\frac{\partial}{\partial x^i}\bigg|\_p$$ precisely? They are usually just called *derivations*, and not much further is explained in most books, but to me they seem like they are functions taking as inputs functions and returning real numbers. If so what is their domain, is it the set of all functions on $\mathbb{R}^n$? 

What I'm basically looking for is a way to make the construction of the basis for the tangent space of $\mathbb{R}^n$ at a point more rigorous, because at the moment it seems very symbolic based on the definition above.