Imagine you got a Heaviside step function $H(x-x_0)$ and the problem requires one to solve the integral $\displaystyle\int_{-\infty}^{\infty} H'(x-x_0) f(x) dx$.
Way 1: One immediately thinks that $H'(x-x_0)$(i.e derivative of step function) is not a function anymore as it's infinite at $x_0$, so it's immediately invalid to put it inside the integral and try to solve it, but by looking at Youtube, the author goes and uses integration by parts and ends up with $f(x_0)$.
Question 1: was his way definitely justified ? meaning that since $H'(x-x_0)$ is not a function anymore on $R$ space, can "integration by parts" be done ? I was told that this can only be justified If $H(x-x_0)$ is treated as a distribution instead of a function. Can you explain why if treated a distribution, then integration by parts is valid and if not treated a distribution, integration by parts is not valid ?
The author comes up with a definition such as he calls $\delta$ to be a distribution $\delta: f \implies \int_{-\infty}^{\infty} \delta(x-x_0)f(x)dx = f(x_0)$.
Way 2: After looking at this topic for so long, I also see the following:
$\delta[f] = \lim_{\varepsilon \to 0}\int \varphi_\varepsilon (x) f(x) dx$ (where $\varphi_\varepsilon$ is a spike function and its area is 1 and whole RHS gives $f(0)$ as well.
Question 2: What's the motivation behind why we use Way 2 ? as in why did we come up with spike function and limit and so on, because integration by parts(way 1) seems to exactly give the same result.
Let's take a step back and recall a few things about distributions and measure theory. Distributions are generalized functions which cannot be integrated in the usual sense. But what does it mean exactly ? That the "standard" integral, i.e. Riemann's integral, which is defined by the limit of the concatenated area of small rectangles of length $\Delta x \rightarrow 0$, is not properly defined. That is why distributions are integrated with respect to other measures instead, meaning that the limit behind $\mathrm{d}x$ is defined differently. For example, the Dirac measure, associated to the Dirac delta function $\delta(x-x_0)$ gives all the weight to the the point $x_0$ and all the other subsets of the domain of integration are ignored, hence $\int_{-\infty}^\infty f(x)\delta(x-x_0) \,\mathrm{d}x = f(x_0)$, because the only (zero-length) rectangle that is taken into account when computing the "Riemann sum" is at $x = x_0$.
This is why we say by abuse of language that $\delta$ is zero everywhere except at one point where it is infinite, so that "$\delta(x-x_0) \cdot \mathrm{d}x = \infty \cdot 0 = 1$" at $x = x_0$ in a way. These oddities are only artifacts when forcing the distribution to behave as a classical Riemann-integrable function in order to sweep all the measure-theoretical stuff, even though it shouldn't. The (initial) purpose of atomic measures such as the Dirac one is the unification of discrete summation and continuous integration as a single object for probability theory. Note that the spike function is a way to "simulate" the Dirac measure by a standard function with a domain of definition reduced to an infinitesimal interval and ultimately to a single point after taking the limit.
After having defined how measures work, all the techniques of integration, such as integration by parts, can be reconstructed and extended to distributions. In the present case, one has : $$ \begin{align} \int_{-\infty}^\infty f(x)H'(x-x_0) \,\mathrm{d}x &= [f(x)H(x-x_0)]|_{-\infty}^\infty + \int_{-\infty}^\infty f'(x)H(x-x_0) \,\mathrm{d}x \\ &= f(\infty) - 0 - \int_{x_0}^\infty f'(x) \,\mathrm{d}x \\ &= f(\infty) - [f(x)]_{x_0}^\infty \\ &= f(x_0) \end{align} $$ It is to be noted that usually the test functions like $f$ are supposed to decrease and vanish at infinity in order to avoid problematic expressions such as $f(\infty) - f(\infty)$.