Basically, I need something which "smooths out" the indicator function of an interval, and which transitions to 0 at a calibratable rate.
The bump function times $e$, as it is, doesn't work, because the transition from 1 to 0 at the boundaries is too gradual. EDIT: I found out that changing the numerator of the argument of the exponential, i.e., going from
so I could as well as close the question now :) However, I'm still curious how this is done usually in (applied?) mathematics. I don't need strictly compact support, i.e., my function needs only to asimptotically vanish, but it doesn't need to be exactly equal to zero anywhere on the real line. Thus the following function works:
Is it usual to use this kind of function to "smooth out" the indicator function, or is it more common to use the bump function? I seem to remember that during university we were shown a lot of functions with a similar graph, so it must be something quite common, but I can't recall their expressions.