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
Famous quotes containing the words smooth, transition and/or functions:
“It may seem strange that any road through such a wilderness should be passable, even in winter, when the snow is three or four feet deep, but at that season, wherever lumbering operations are actively carried on, teams are continually passing on the single track, and it becomes as smooth almost as a railway.”
—Henry David Thoreau (1817–1862)
“When I was going through my transition of being famous, I tried to ask God why was I here? what was my purpose? Surely, it wasn’t just to win three gold medals. There has to be more to this life than that.”
—Wilma Rudolph (1940–1994)
“The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
—Cyril Connolly (1903–1974)