I just dabbled into the field of generalized functions (or theory of distributions). I am a bit confused as to how the class of generalized functions includes that of ordinary functions.
Let $K$ be the space of all test functions; let $f$ be a function in the ordinary sense; let $\int f \phi$ exist for all $\phi \in K$. (Please also tell me if my understanding of the setting goes wrong.) My initial guess is that, since there is a bijection such that $f \mapsto \{ \int f \phi \}_{\phi \in K}$ and vice versa for all suitable $f$ and all regular generalized functions $\{ \int f \phi \}_{\phi \in K}$, we can identify an ordinary function $f$ with a generalized function $\{ \int f \phi \}_{\phi \in K}$. Is this correct? Thanks.
Consider for example a set of functions that has been used in physics: The set $F$ of infinitely differentiable $f:\Bbb R \to \Bbb R$ such that $f(x)$ and each of its derivatives converges uniformly to $0$ as $|x|\to \infty,$ and such that $\int_{\Bbb R} f^{(j)}(x)dx$ exists for all $j$.... where $f^{(0)}=f,$ and $f^{(j)}$ is the $j$-th derivative of $f$ when $j>0.$
Now let $B[\Bbb R]$ be the set of bounded integrable $g:\Bbb R\to \Bbb R.$ For $g\in B[\Bbb R]$ and $f\in F$ define $g^*(f)=\int_{\Bbb R}f(x)g(x)dx.$ Then $g^*:F\to \Bbb R$ is linear , that is $g^*(f_1+kf_2)=g^*(f_1)+kg^*(f_2)$ for $f_1,f_2 \in F$ and $k\in \Bbb R.$
But there are linear maps from $F$ to $\Bbb R$ that are not in the set $B^*=\{g^*: g\in B[\Bbb R]\}.$ For example, $\delta(f)=f(0)$ for all $f\in F.$
However $\delta$ is the point-wise limit of a sequence of members of $B^*.$ That is, there is a sequence $(g_n)_{n\in \Bbb N}$ in $B[\Bbb R]$ such that $\lim_{n\to \infty} g^*_n(f)=\delta (f)=f(0)$ for each $f\in F.$
So we may consider a generalized function to be a linear map $h:F\to \Bbb R.$ A feature of the Heaviside operational calculus is that we can often treat such $h$ as if they were members of $B^*,$ even as if they of the form $g^* $ with continuously differentiable $g\in B[\Bbb R]$ when in fact they are point-wise limits (as in the sense above) of sequences of such $g^* .$
I hope this clarifies some of the issues. It can be confusing to read of a function $\delta$ such that $\delta (x)=0$ when $x\ne 0,$ but $\int_{\Bbb R}\delta(x)dx=1.$