Non-analytic Smooth Function - Smooth Transition Functions

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:

    The moderation of people in prosperity is the effect of a smooth and composed temper, owing to the calm of their good fortune.
    François, Duc De La Rochefoucauld (1613–1680)

    A transition from an author’s books to his conversation, is too often like an entrance into a large city, after a distant prospect. Remotely, we see nothing but spires of temples, and turrets of palaces, and imagine it the residence of splendor, grandeur, and magnificence; but, when we have passed the gates, we find it perplexed with narrow passages, disgraced with despicable cottages, embarrassed with obstructions, and clouded with smoke.
    Samuel Johnson (1709–1784)

    The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.
    Ralph Waldo Emerson (1803–1882)