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
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