How to precisely define $C^\infty$ in $f(x) \in C^\infty$ In single variable calculus, a common way to denote a function that is continuous for all derivatives is to write $f(x) \in C^\infty$ i.e. $f(x) = \exp(x)$
Is there a more rigorous way to define $C^\infty$ (using set notations, so forth such that it can be generalized to higher dimensions)?
 A: Let $C^0$ be the set of continuous functions.  Then
$$C^\infty = \{ f | \:\forall n\:\: \frac{d^nf}{dx^n} \in C^0\}$$
A: For the one-dimensional case it is actually not very difficult.
Assume $I\in \Bbb R$ is an open connected subset of real line. Then denote $C(I) = C^0(I)$ the set of all functions $\ f:I\to\Bbb R$ continuous on $I$, i.e.
$$
C^0 = \left\{ \ f: I \to \Bbb R \ \Big| \ \ \forall \, x_0 \in I \quad  \lim_{x \to x_0} f(x) = f\!\left(x_0\right)   \right\}
$$
Then denote $C^1(I)$ the set of (once) continuously differentiable functions : 
$$
C^1 = \left\{ \ f: I \to \Bbb R \ \Big| \ \ \forall \, x_0 \in I \quad \exists \ \ f' \!\left( x_0\right) = \lim_{\Delta x \to 0} \; \frac{f\left(x_0 + \Delta x \right) - f\left( x_0\right) }{\Delta x}, \ \text{ and } \ f' \in C^{0}(I) \right\}.
$$
By induction we can define  the set $C^n(I)$ of $n$-times continuously differentiable functions:
$$
C^n = \left\{ \, f: I \to \Bbb R \, \Big| \ \forall \, x_0 \in I \  \exists \ \ f^{(n)} \!\left( x_0\right) = \lim_{\Delta x \to 0} \frac{f^{(n-1)}\left(x_0 + \Delta x \right) - f^{(n-1)}\left( x_0\right) }{\Delta x}, \text{ and } f^{(n)} \in C^{0}(I) \right\},
$$
where $f^{(n)}$ is the $n$-th derivative of $f$ defined as 
$$
\begin{cases}
\displaystyle{ f^{(n)} \!\left( x_0\right) = \lim_{\Delta x \to 0} \frac{f^{(n-1)}\left(x_0 + \Delta x \right) - f^{(n-1)}\left( x_0\right) }{\Delta x}}, & n\ge 2
\\
\displaystyle{ f^{(1)}\!\left( x_0\right)  = f' \!\left( x_0\right) = \lim_{\Delta x \to 0} \; \frac{f\left(x_0 + \Delta x \right) - f\left( x_0\right) }{\Delta x}}, & n = 1.
\end{cases}
$$  
Finally, we define
$$
C^\infty(I) = \left\{ \ f: I \to \Bbb R \ \Big| \ \ f \in C^n (I) \quad \forall \, n \in \Bbb N^+ \right\}.
$$
