This is a short note on creating bump functions, test functions which are on the desired domain, etc. I will be working in one dimension. However, all these results can be generalized to higher dimensions by using polar coordinates.
As we know, the function for and for is a smooth function. Hence, it is an ideal candidate for constructing smooth, compactly supported functions. If we wanted to construct a smooth function that was supported on , then is one such function.
However, the main difficulty is in constructing a bump function of the desired shape. How do we construct a bump function that is on ? The idea that I had, which is different from the literature that I’ve consulted (including Lee’s “Smooth Manifolds”), is that we could consider the integrals of functions.
Basically, it we are integrating a function that is positive on , and then adding that to the integral of the negative of the same function, but now supported on .
This function will be constant on , and then decrease to on . On re-scaling (multiplying by a constant), we can obtain a bump function on that is on .