The author of a book I'm reading at a certain point says:
Let $f:\mathbb R\rightarrow \mathbb R$ be a $C^\infty$ function satisfying the conditions
- $f(0)> \epsilon$ where $\epsilon$ is a real number greater than $0$
- $f(r)=0\quad \quad \quad \quad$ for $ r\geq 2\epsilon$
- $-1<f'(r)\leq 0\quad$ for all $r\in \mathbb R$
Intuitively it convinces me, but I'd like to see a proof of this fact: an example of a a function like that or some results which show it exists. Is there any standard way to approach problems of this kind? Otherwise, could someone give me some references to look at?
Thanks
For $p>0,$ define
$$g_p(x) = \begin{cases} \exp\left (\frac {-p}{x(2-x)}\right)& 0<x<2 \\0 &\text { elsewhere } \end{cases}.$$
Then $g_p\in C^\infty(\mathbb R).$ Note that
$$\int_{\mathbb R} g_p = \int_0^2 g_p <2$$
for all $p>0.$ As $p \to 0^+,$ $\int_0^2 g_p \to 2,$ and as $p\to \infty,$ $\int_0^2 g_p \to 0.$ Because this integral is a continuous function of $p,$ there exists $p_0>0$ such that $\int_0^2 g_{p_0} = 1.$ (You can use the dominated convergence theorem to verify these statements.)
Now define
$$f(x) = 1 - \int_{-\infty}^x g_{p_0}.$$
This function does what you want, for the special value $\epsilon = 1.$ The function $\epsilon f(x/\epsilon)$ will then handle the general situation.