- #1

Eclair_de_XII

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- TL;DR Summary
- The function of continuity is paraphrased as follows.

A function ##f## is said to be continuous at some point ##x_0## in some field if for all positive values of ##\epsilon##, there is an integer ##N## with the property that whenever a natural number ##n## is greater than or equal to ##N##, ##f(x_n)## is within distance ##\epsilon## of ##f(x_0)##, where ##x_n## is a sequence of points converging to ##x_0##.

I argue not. Let ##f:\mathbb{Q}\rightarrow\mathbb{R}## be defined s.t. ##f(r)=r^2##. Consider an increasing sequence of points, to be denoted as ##r_n##, that converges to ##\sqrt2##. It should be clear that ##\sqrt2\equiv\sup\{r_n\}_{n\in\mathbb{N}}##. Continuity defined in terms of sequences of numbers requires that whenever a sequence of points converges to some point, the image of the sequence must converge to the image of that point, also. However, since the concept of the supremum does not even exist in the field of rational numbers, the definition given fails, since the convergence of the sequence given depends on aforementioned concept.