What is the definition of an ordinary smooth function?

Question

Can't find it anywhere. According to the textbook, PDE's by olver, the dirac delta function represents a unit impulse concentrated along a single point, and is a limit of a sequence of ordinary smooth functions which represent progressively more and more concentrated unit forces. These are the two properties of the delta function, $\delta _{\xi}(x)=0$ for $x\neq \xi$ and $\int_{a}^{b} \delta _{\xi}(x)=1$ for $a<\xi <b$, and the limit function also satisfies the two properties given that $\lim_{n\to\infty}g_{n}(x)=\delta _{\xi}(x)$. This characterization of the impulse force should help in solving linear boundary value problems governed by ODE's on an interval $a<x<b$

Answer 1

A function $f$ is an ordinary smooth function if $f'$ is defined and continuous on the domain.

Answer 2

Smooth usually means that all derivatives of the function exist and are continuous at all points in the domain (though sometimes definitions vary). Ordinary has no technical meaning here. This line in the book is not meant to be technical--I will attempt below to explain what Peter is getting at.

The "delta function" is not a function in the usual sense (i.e., a rule assigning to each $x$ a real number $f(x)$). If it were, then any reasonable theory of integration (e.g., Riemann or Lebesgue integral) would require the integral of the delta function to be zero (since it is zero at all but one point). The delta function is a "generalized function" or "distribution", and is defined by how it acts on other functions.

The point Peter is making is that even though the delta function is this strange generalized function, it is in fact not too strange because you can express it as the limit (in some sense) of ordinary functions. By ordinary, we mean a rule assigning to each $x$ a real number $f(x)$, so not a distribution or generalized function. The sequence of functions should integrate to 1, and become more and more concentrated around the origin in the limit. There are an infinity of ways of expressing the delta function as the limit of a sequence of ordinary functions. In particular the functions need not be smooth. For example, the sequence of discontinuous functions

$$f_n(x)=\begin{cases} n,&\text{if }0 \leq x \leq \frac{1}{n}\\ 0,&\text{otherwise}\end{cases}$$

converges to the delta function. Hope this helps.