this from +michael livshits
a gem from a comment. Need repost to study later.

it's about differentiation without delta epsilon.


+Xah Lee​​​​​​​​, You may like so-called automatic differentiation if you want to get away from epsilons and deltas, check it out on wikipedia. Also differentiation can be viewed as factoring f(x)-f(a) through x-a in a certain class of functions, for polynomials it is explicit, the most immediate generalization is to assume that the functions involved are Lipschitz, it takes care of all functions with Lipschitz derivative, and the definition of the derivative becomes the inequality
|f(x)-f(a)-f'(a)(x-a)≤K(x-a)^2 with some constant K, uniform in x and a. See Karcher's lectures and my article at http://www.mathfoolery.com/Article/simpcalc-v1.pdf for details. To prove differentiability in this context is to check this specific inequality. Likewise, to check continuity it's enough to check the inequality |f(x)-f(a)|≤Km(|x-a|), uniform in x and a, where m is a modulus of continuity and K is a constant, any continuous function on a closed bounded set has a modulus of continuity. These notions are widely used in PDEs, harmonic analysis and function theory, where we deal with spaces of functions with certain modulus of continuity (Lipscitz, Holder, etc.) instead of dealing with all the continuous functions at once.

The bottom line is that the elementary calculus can be greatly simplified and made more widely accessible and understandable if we focus on reasonable functions instead of slavishly following the orthodox classical way of defining pointwise continuity and differentiability by epsilon-delta and then using compactness to get any useful or applicable results.

Actually most of just continuous functions are highly pathological, the derivatives of pointwise differentiable functions are even more pathological. As Peter Lax said, no respectful analyst will talk about the space of pointwise differentiable functions. But then again, there is Henstock-Kurzweil integral that can integrate any pointwise derivative and get the fundamental theorem of calculus without any restrictions on the derivative, but it heavily relies on compactness and is hardly appropriate for elementary calculus.

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