All of these involve breaking up the domain by inequalities and, whether visible or not, involve something no simpler that the exponential function. The original one-sided version is $ f(x) = e^{-1/x} \; \mbox{for} \; x > 0 $ but $ f(x) = 0 \; \mbox{for} \; x \leq 0. $
You can get a bump from this by multiplication with $ g(x) = f( 1 + x) \cdot f(1 - x) $
You get a smoothed step function from $ h(x) = \int_{- \infty}^x \; g(t) dt $
You get a plateau bump function, constant in the middle, from $ p(x) = h(x + A) \cdot h(A -x) $ for some $A > 1.$
We can prove some properties of this sort of thing. It has no removable singularity at the points where it is not real analytic, at best it has an essential singularity or possibly is not even defined in any neighborhood of the point in $\mathbb C.$