Continuity
A real function f is continuous at a standard real number x if for every hyperreal x' infinitely close to x, the value f(x' ) is also infinitely close to f(x). This captures Cauchy's definition of continuity.
Here to be precise, f would have to be replaced by its natural hyperreal extension usually denoted f* (see discussion of Transfer principle in main article at non-standard analysis).
Using the notation for the relation of being infinitely close as above, the definition can be extended to arbitrary (standard or non-standard) points as follows:
A function f is microcontinuous at x if whenever, one has
Here the point x' is assumed to be in the domain of (the natural extension of) f.
The above requires fewer quantifiers than the (ε, δ)-definition familiar from standard elementary calculus:
f is continuous at x if for every ε > 0, there exists a δ > 0 such that for every x', whenever |x − x' | < δ, one has |ƒ(x) − ƒ(x' )| < ε.
Read more about this topic: Non-standard Calculus
Famous quotes containing the word continuity:
“Continuous eloquence wearies.... Grandeur must be abandoned to be appreciated. Continuity in everything is unpleasant. Cold is agreeable, that we may get warm.”
—Blaise Pascal (1623–1662)
“Only the family, society’s smallest unit, can change and yet maintain enough continuity to rear children who will not be “strangers in a strange land,” who will be rooted firmly enough to grow and adapt.”
—Salvador Minuchin (20th century)
“If you associate enough with older people who do enjoy their lives, who are not stored away in any golden ghettos, you will gain a sense of continuity and of the possibility for a full life.”
—Margaret Mead (1901–1978)