As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let it go to zero in the end: $$\int^{+\infty}_{-\infty} dx e^{ikx -\epsilon|x|}= 2\pi\delta(k).$$
We may use the definition $$\delta(x) = \frac{1}{\pi}\lim_{\epsilon\rightarrow 0^+} \frac{\epsilon}{x^2+\epsilon^2}$$ and various other representations of the Dirac delta function to write down the transform above.
A related one is that, in physics, when we need to calculate the Fourier transform of a "misbehaved function multiplied by a step function $\theta(x)$, we add an infinitesimal number to make it converge: $$\int^{+\infty}_0 f(x) e^{ikx-\eta^+ x} dx.$$ Another example is, when we define the operation of time-ordering operator, we always define the equal time Green's function to be $$G(t,t) \equiv - \langle T_t c(t)c^{\dagger}(t) \rangle \equiv -\langle T_t c(t)c^{\dagger}(t+0^+)\rangle,$$ where $c/c^{\dagger}$ is a fermion creation/annihilation operator.
Is this kind of "infinitesimal techniques" all rigorously defined, for example, some in the language of distributions?
I apologise for this very general question. I cannot of course say what "all" is. I do not expect this to be a good question, but still I am curious about mathematicians' point of view on that if they ever know this situation in physics.
The first two examples you mention appear respectively in Fourier and Cauchy who specifically mention that the epsilon is infinitesimal. Of course they didn't have distribution theory. Nor is distribution theory necessary to make this work; one can use a modern theory of infinitesimals where one can define Dirac delta functions possessing local values (of course on the extended continuum). The idea goes back to Robinson's 1966 book. Some of the relevant literature can be found in this article.