Why Is The Squaring Function Not Uniformly Continuous?
Let f(x) = x2 defined on . Let be an infinite hyperreal. The hyperreal number is infinitely close to N. Meanwhile, the difference
is not infinitesimal. Therefore f* fails to be microcontinuous at N. Thus, the squaring function is not uniformly continuous, according to the definition in uniform continuity above.
A similar proof may be given in the standard setting (Fitzpatrick 2006, Example 3.15).
Read more about this topic: Non-standard Calculus
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