I am looking for a function $f:\mathbb{R} \to\mathbb{R}$ that is $C^\infty$, that equals zero when $|x| \geq 1$, that is strictly positive when $|x|< 1$ and that is constructively defined. Those functions are heavily used in differential geometry and distributions for example.
The usual definition of bump function $g(x) = \exp(-1/(1-x^2))$ for $|x| < 1$ is not constructive, because the comparison of an arbitrary real number $x$ to 1 is not computable. This definition can be improved, using the fact that the absolute value of real numbers is constructively defined (a real number is a Cauchy sequence of rational numbers, and its absolute value is the Cauchy sequence of the absolute values). So we get this bump function $$ g_1(x) = (1-x^2+|1-x^2|)\exp(-1/(1-x^2))$$ But that is still not constructive, because $1/(1-x^2)$ is not directly defined at $\pm 1$, $g_1$ is extended by continuity there. That still needs to compare $x$ to 1.
