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.

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.

View 6 previous comments

- +Xah Lee Yes, you got it, I have something against the math establishment and math tradition (and something else too), specifically, I have a lot against the calculus sham :-)52w
- +Xah Lee For the rest of it ask Dick :-)52w
- +Xah Lee See what happens to these epsilon-delta and reduce-everything-to-limits advocates? They realize that their position is indefensible and go back to their woodwork :-)51w
- +Xah Lee Ask Karcher what he thinks about calculus, he will give you an earful :-)51w
- +michael livshits your statement about john is not fair. You presented your views in a attacking style, repeatedly many times in different threads. And as far as i know, john never focused or emphasized the math foundation issues. So, in a sense, you imagined a wrong enemy.

and john baez is a friend. Via his blogs and help, i learned a lot math, and never seen slight hostility towards others.

and originally this thread could be just about math, but now it's something else. Not good for me neither.

am myself kinda oddball and have offended many in programing community... so i kinda have a heart with non-conformists or non-mainstream. But i think let's not get into that. Or, it should be discussed in another thread or privately.51w - +Xah Lee Why not get into that, math teaching and math are intimately related. As for John Baez, I have nothing against him personally, it's just some of his opinions that I strongly disagree with :-)51w

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