Smooth Transition Functions
The function
has a strictly positive denominator everywhere on the real line, hence g is also smooth. Furthermore, g(x) = 0 for x ≤ 0 and g(x) = 1 for x ≥ 1, hence it provides a smooth transition from the level 0 to the level 1 in the unit interval . To have the smooth transition in the real interval with a < b, consider the function
For real numbers a < b < c < d, the smooth function
equals 1 on the closed interval and vanishes outside the open interval (a,d).
Read more about this topic: Non-analytic Smooth Function
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