smooth bump function

Existence of bump function

Given any real numbers $0\le r1<r2$, there exists a smooth function $f:{\mathbb{R}}^n\to \mathbb{R}$ such that $f(x)=1$ for $x\in B_{r_1}(0)$ and $f(x)=0$ for $||x||>r_2$.

Proof

We can take the smooth function

$$f(x)=\{\begin{array}{ll}{e}^{-1/t}& t>0\\ 0& t\le 0\end{array}$$

Then we can make a smooth cutoff function

$$F(x)=\frac{f(b-x)}{f(b-x)+f(x-b)}$$

This has the property that $F(x)=1$ for $x<a$ and $F(x)=0$ for $x>b$.

Then, if we want to make the bump function as desired above we can take the function

$$\phi (x^1,\dots ,x^n)=F\left(\sqrt{(x^1{)}^2+\dots (x^n{)}^2)}\right)$$

which satisfies the requirements and is smooth.