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.

2 fixed typo

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 $2\pi/3$ 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 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. 1 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$2\pi/3$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 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.