# Writing a two variable function$f(x,t)$ in terms of Dirac-Delta $δ(x)$ function and a function $P(t)$?

How to write a two variable function $$f(x,t)$$ in terms of Dirac-Delta $$\delta(x)$$ function and a function $$P(t)$$?

For example;

I read something in a book. You can find the following picture.

But I don' t understand the logic behind this. Could you explain it?

$$W=\int_1^2\boldsymbol{F}\cdot d\boldsymbol{r}$$
In this case, that would mean that all the force is acting on a single point. An integral over a single point gives $$0$$ ($$\int_1^1 = 0$$). But since we do want the force to act on the object and do work, we don't want this. In order to have a finite integral value one need to "infinitely concentrate" the force on that point. Thus, the delta dirac distribution.