Find a smooth function $\eta:\mathbb{C}\to\mathbb{R}$ whose support is a disk.

Question

This comes from the proof of the following lemma in Jost's Compact Riemann Surfaces (Lemma 2.3.3).

Lemma 2.3.3 Every compact Riemann surface $\Sigma$ admits a conformal Riemann metric.

proof. ... For a disk $D\subset\mathbb C$ we choose a smooth function $\eta:\mathbb C\to\mathbb R$ with $$\eta>0\text{ on }D,\quad\eta=0\text{ on }\mathbb C\backslash D$$ ...

My questions:

(1) Does "smooth" here mean "infinitely differentiable as a $\mathbb R^2\to\mathbb R$ function (just to make sure)?

(2) How to guarantee the existence of such functions?

For (2) I know such functions must be smooth but non-analytic. The only example I know is $$f(x)=\left\{\begin{array}{lll}e^{-1/x}&,&x>0\\0&,&x\leq0\end{array}\right.$$ But how to generalize this to an $\mathbb R^2\to\mathbb R$ function?

Answer 1

$f(x)=e^{-\frac 1 {1-x}}$ for $x<1$ and $0$ for $x \geq 1$ defines a smooth function which is positive on $(-\infty,1)$ and $0$ outside it. So $f(\|x\|^{2})$ is a smooth function on $\mathbb R^{2}$ which is positive for $\|x\|<1$ and $0$ elsewhere. For any other disk in $\mathbb R^{2}$ use an appropriate affine transformation.

Answer 2

Define$$\eta(z)=\begin{cases}e^{\frac1{\lvert z\rvert^2-1}}&\text{ if }\lvert z\rvert<1\\0&\text{ otherwise.}\end{cases}$$