> I think the interesting thing here is that with relatively little training you can start to compute with these numbers, which is definitely not the case with analysis on distributions.

I don’t know, this feels like a math “hold my beer” moment. Math is infinitely deep and interconnected, but you have to start somewhere, on solid ground.

I was not being facetious above - the issues that i mentioned above are actual problems when you make calculations. But let’s ignore those issues for a second.

So you found the “derivative” of a single, arbitrary chosen representative of an infinite family of functions. What if you chose (tanh(Nx)+1)/2? What if you chose Logistic(N^2 x) instead of Logistic(N x)? You’d get different derivatives. In fact any function (up to additive constant) whose integral of the neighborhood of 0 is 1 would work there. What use are the values you are calculating if they reflect your choice and not anything inherent to the problem?

As for distributions, i picked up and read a small 100 page penguin “leaflet” from the library during my undergrad that went through the subject rigorously (and with plenty of examples). It’s not that different from working rigorously with probability or real analysis. And at the end, in applications we indeed are usually interested in integrals, not derivatives which we have not even defined. At the end of the day, you have a [X=weak L^infinity(R)] function (heavyside). You look at the dual space and since we established don't really need the deep theory, believe me when i tell you that the correct space is the space of test function on R (X’=infinitely smooth, compact support, bounded integral). Each of those conditions is simple for our simple example of R. The inner product is via integral.

Formally speaking elements of X are equivalence classes of sequences of functions and are not really defined pointwise, but neither was the NSA example. There we had to choose an arbitrary representative hyperreal function and here we may identify pointwise defined functions with the classes of the constant sequences of those functions.

using integration by parts it is simple to show that <F,G’> = <F’,G> if F is continuously differentiable on G’s support. Let us formally define in this way the weak derivative for functions that are not traditionally differentiable, if such an element exists an is unique that satisfies all the integral relations. However note that differentiation is an linear isomorphism on the space of test functions and so weak derivative indeed exists and is unique. Furthermore

We can also define elements of X poinwise by identifying F(x) with the limit <Txn,F> as n grows if it exists and is independent of the sequence Txn where Txn is a sequence of functions with support tending to {x} and constant integral 1. It is a simple exercise to show that for “normal” functions this holds, and by above we can poinwise define derivatives this way as well.

What about our H(x)? it is an exercise to check that by pointwise we get what we should outside of 0. What about the derivative at 0? Well, do the exercise above with <T0n’,H> and we see that it is penrose undefined. Decidedly not even necessarily infinite, just undefined. However, integration by parts shows that <T,DH>=T(0) ie dirac delta at 0.

Aside from all the theory that i kinda gave handwavingly much like OP in the post, the mechanics are simple integration by parts to get the only stuff that’s “real” here, which are the integrals. in NSA we haven’t even defined those. How will knowing what infinity i will get at 0 given an arbitrarily chosen representative for H help me?

Do your results depend on ZFC? stronger axioms? At what level of infinity do we stop? You can brush aside the formalities but then what better is this approach than physicists?