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:
“he changed and ran
Through many shapes; I lunged at the smooth throat
Of a great eel; it changed, and I but smote
A fir-tree roaring in its leafless top;
And thereupon I drew the livid chop
Of a drowned dripping body to my breast....”
—William Butler Yeats (1865–1939)
“The god or hero of the sculptor is always represented in a transition from that which is representable to the senses, to that which is not.”
—Ralph Waldo Emerson (1803–1882)
“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (1803–1882)