Show that $\phi_{y,\epsilon} \in D(\mathbb{R^m})$

Question

For a fixed $y$

$$\phi_{y,\epsilon}(x)= \left\{ \begin{array}{ll} \exp(-\frac{\epsilon^2}{\epsilon^2-|x-y|^2}) & \mbox{if $|x-y| \lt \epsilon$};\\ 0 & \mbox{otherwise}.\end{array} \right. $$

Show that $\phi_{y,\epsilon} \in D(\mathbb{R^m})$

I am unable to show this. Thanks for the help!!

Answer 1

Show by induction that all derivatives of $f(t)=\exp(-1/t)$ are of the form $f^{(n)}(t)=p_n(1/t)f(t)$ with polynomials $p_n$.Then conclude that $f$ extended by $0$ for $t\le 0$ is a smooth function and note that your function is a composition of $f$ with a polynomial.