Reference
OpenAI's Answer
Non-standard analysis describes the continuity of a function using hyperreal numbers through the concept of infinitesimals. Here's a brief explanation:
In non-standard analysis, the real number system is extended to include hyperreals, which consist of the usual real numbers along with infinitesimally small and infinitely large numbers. Continuity at a point can be defined using hyperreals as follows:
A function ( f ) is continuous at a point ( a ) if, whenever ( x ) is infinitesimally close to ( a ) (denoted ( x \approx a )), then ( f(x) \approx f(a) ). This means that any hyperreal ( x ) that is infinitely close to ( a ) results in ( f(x) ) being infinitely close to ( f(a) ).
This approach provides an intuitive framework for understanding continuity without the need for traditional (\epsilon)-(\delta) proofs.
For more details, you can refer to these resources:
Additionally, you might find the visual illustration of hyperreals useful:
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