Consider

The solution is given by $u(x,t)=f(x+ct)+g(x-ct)$ by method of characteristics with $u\in C^2(\mathbb{R})$ but by looking at the form of the solution $u$ we see that from the solution there is no apparent solution why such regularity is needed. (A contrast to the solutions of the Burger's equation which forms singularities).

Consider the Heaviside function $H:\mathbb{R}\to \mathbb{R}$ given by:

Classical derivative has a problem at $x=0$. A more general derivative which we shall call $\delta$ function. However notice that:

This cannot hold for any 'ordinary' function. Moreover, Dirac wants to also know the derivatives of this $\delta$.
$\delta$ will be defined as a distribution or so called "generalized function".

A distribution is a function $T$ that takes any test function, $\varphi :\mathbb{R}\to \mathbb{R}$ and maps it to ${\int }_{\mathbb{R}}T(x)\varphi (x)dx$.

For any real/complex valued function f defined on $\mathrm{\Omega }\subset {\mathbb{R}}^n$, the support of $f$ is

If $\mathrm{\Omega }$ is any open set on ${\mathbb{R}}^n$, the space of test function on $\mathrm{\Omega }$ is:

If ${\varphi }_n\in D(\mathrm{\Omega })$ we say ${\varphi }_n\to 0$ in $D(\mathrm{\Omega })$ if:

1. There exists a compact set $K\subset \mathrm{\Omega }$ such that $\text{supp }{\varphi }_n\subset K\mathrm{\forall }n$.
2. $\underset{n\to \mathrm{\infty }}{lim}\underset{x\in \mathrm{\Omega }}{\text{max}}|D^{\alpha }{\varphi }_n(x)|=0$ for any multi index $\alpha$

A linear mapping $T:D(\mathrm{\Omega })\to \mathbb{C}$ is called a distribution if $T({\varphi }_n)\to T(\varphi )$ whenever ${\varphi }_n\to \varphi$.
The set of all distributions on $D(\mathrm{\Omega })$ is denoted by $D^{\prime }(\mathrm{\Omega })$.

We say that two distributions $T_1$ and $T_2$ are equal if: