In cylindrical coordinates $x=\{r,\theta,z\}$, let the axi-symmetrical scalar function $f(t,r,z)$ be $C^1$ in $t\in [0,\infty)=\mathbb{R}_0^+$, smooth ($C^\infty$) in $r\in [0,\infty)=\mathbb{R}_0^+$, and $C^2$ in $z\in \mathbb{R}$.
What is the symbol of the set that $f(t,r,z)$ belongs to?
Can we write as follows? $$f(t,r,z)\in C^1(\mathbb{R}_0^+)C^\infty(\mathbb{R}_0^+)C^2(\mathbb{R})\tag{1}$$ or $$f(t,r,z)\in C^1(t\in\mathbb{R}_0^+)C^\infty(r\in\mathbb{R}_0^+)C^2(z\in\mathbb{R})\tag{1}$$