This is not really an answer, but rather a puzzling example about the notion of smoothness.
Consider the unit disk
$$ D=\lbrace z\in\mathbb{C};\;\;|z|\leq 1\rbrace.$$
The cyclic group $C_n:=\mathbb{Z}/n\mathbb{Z}$, $n\geq 3$, acts on $D$ by rotations of angle $2\pi/n$ about the origin. Consider the quotient $D/C_n$. The functions on this quotient can be identified with the $C_n$-invariant functions on $D$. There are several rings of functions on $D/C_n$
$$ C^0(D)^{C_n} :=\mbox{ continuous, complex valued $C_n$-invariant functions on $D$}$$
$$ C^\infty(D)^{C_n} :=\mbox{smooth, complex valued $C_n$-invariant functions on $D$}$$
$$ \mathcal{H}(D)^{C_n}:=\mbox{holomorphic valued $C_n$-invariant functions on $D$}$$
Here is the surprise. The ring $C^0(D)^{C_n}$ is isomorphic to the ring $C^0(D)$ of continuous complex valued functions on $D$. The reason is that the spaces $D$ and $D/C_n$ are homeomorphic compact spaces and thus their rings of continuous complex valued functions are isomorphic.
Now observe that the ring $\mathcal{H}(D)^{C_n}$ is also isomorphic to the ring $\mathcal{H}(D)$. Indeed, a holomorphic function
$$ f(z)=\sum_{k\geq 0} a_k z^k $$
is $C_n$ invariant iff $a_k=0$, $\forall k\not\equiv 0 \bmod n$. Thus
$$ f(z)\in \mathcal{H}(D)^{C_n} \Longleftrightarrow f(z)=\sum_{k\geq 0} a_{kn} z^{kn} $$
The map
$$ \mathcal{H}(D)\ni f(z) \mapsto f(z^n)\in \mathcal{H}(D)^{C_n} $$
is the sought for isomorphism. These two examples show that we cannot distinguish between $D$ and $D/C_n$ topologically or holomorphically. Surprisingly
$$ C^\infty(D)^{C_n} \not\cong C^\infty(D)$$
This is not obvious but not terribly hard to prove. The upshot of this last fact is that smoothly the disk $D$ and the cone $D/C_n$ are different.
The point of this simple example is that smoothness is a rather subtle concept. More subtle examples can be found in Kolmogorov's work on Hilbert's 13th problem. Using probabilistic ideas he gives a precise quatitative quantitative meaning to the fact that smooth functions are fewer than continuous functions.
MR0112032 (22 #2890) Kolmogorov, A. N.; Tihomirov, V. M. ε-entropy and ε-capacity of sets in function spaces. (Russian) Uspehi Mat. Nauk 14 1959 no. 2 (86), 3–86.
Vituškin, A. G.; Henkin, G. M. Linear superpositions of functions. Uspehi Mat. Nauk 22 1967 no. 1 (133), 77–124.
