The reals you only get once you go beyond algebra and arithmetic and take the closure under convergence of Cauchy sequences.

]]>in order to have a perfectly closed field for all arithmetic operations;

totally trivial, like the first trigonal number, 1(1+1)/2 ]]>

Rookie mistake; I’ll fix it shortly ðŸ˜€

]]>“in that in order for a function to be differentiable at some point p, the function will also need to be differentiable at every point in some interval around p.”

Of course the functions people most often use will have nice properties like that, but it turns out that this does not follow from definitions. An example of a function differentiable at only one point is a function f such that it is x^2 at rational numbers and -x^2 at irrationals. (if we have not defined irrationals yet we can make a function in the same way that depends on (for example) the denominator of your rational or what have you.)

And while you talk about continuity, it turns out that this statement is still not true for all continuous functions, but again, only nice ones. For example, starting with the Weierstrass function, call it W(x), in the same spirit as the example above, define f(x) = x^2 * W(x), which will be continuous everywhere but differentiable only at x=0.

I am not sure of what exact conditions are needed to make that statement true, just remembered some examples of exotic functions from my calc classes. Anyway I thought it was something you should fix. ]]>

and p00r j.g just knows not a thing about the theory of numbers,

that is to say modular arithmetic since Fermat and his “last” theorem …

which clearly was one of his first insights into what are now known as p-adics, but,

hey! ]]>

(EDIT: I updated the post accordingly) ]]>

People like Wildberger might point out (incorrectly) that you’re implicitly using Real numbers when you let your arbitrary epsilon be a “number” greater than zero. Of course, you did say in the beginning that we’re working only with N and Q right, and at the end that we haven’t even talked about the reals yet.

Still, I think instead of saying epsilon is an arbitrary small number, you should write explicitly that it is an arbitrary small rational number. This way you can avoid having cranks repeat such a redundant comment here. ]]>

However I am having issues with your RSS. I don’t understand

why I cannot subscribe to it. Is there anybody else having identical RSS problems?

Anyone who knows the solution will you kindly respond? Thanks!! ]]>