Definition For Functions On Metric Spaces
Given metric spaces (X, d1) and (Y, d2), a function f : X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d1(x, y) < δ, we have that d2(f(x), f(y)) < ε.
If X and Y are subsets of the real numbers, d1 and d2 can be the standard Euclidean norm, || · ||, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all x, y ∈ X, |x − y| < δ implies |f(x) − f(y)| < ε.
The difference between being uniformly continuous, and simply being continuous at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain.
Read more about this topic: Uniform Continuity
Famous quotes containing the words definition, functions and/or spaces:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
—Edward T. Hall (b. 1914)
“Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.”
—Blaise Pascal (1623–1662)