Delta function as a probability density distribution? What does it mean to define a probability density function as Dirac-delta function. I mean Dirac delta is not a function itself. So how is it considered to be an example of a density function?
I need an exact mathematical definition please. Because in the literature that I am studying it has been used without referring that it might be a problem at all.
 A: The relevant mathematical notion is the notion of measure. A measure $\mu$ is defined on a class of subsets of a space $\Omega$, that is, for some subsets $A\subseteq\Omega$, $\mu(A)$ is a nonnegative real number.
The Dirac measure at $x$, usually denoted $\delta_x$, is the unique measure such that $\delta_x(A)=1$ if $x$ is in $A$ and $\delta_x(A)=0$ otherwise. Note that no Dirac measure has a probability density function. As you noted yourself, when $\Omega=\mathbb R$, for every integrable function $f$, one knows that $\int\limits_{x-\varepsilon}^{x+\varepsilon}f\to0$ when $\varepsilon\to0$ hence one cannot expect to get anything else that $\mu(\{x\})=0$ if $\mu$ has density $f$. By contrast, $\delta_x(\{x\})=1$.
The other interpretations are mostly physics; but beware that physicists are able to manipulate non rigorously defined objects in ways mathematicians cannot.
Now, the rigorous definition when the space is $\Omega=\mathbb R$ endowed with the Borel sigma-algebra $\mathcal B(\mathbb R)$ (we already explained nearly everything). Then $\delta_x:\mathcal B(\mathbb R)\to\{0,1\}$ is defined by $\delta_x(A)=1$ for every $A$ in $\mathcal B(\mathbb R)$ such that $x$ is in $A$ and $\delta_x(A)=0$ for every $A$ in $\mathcal B(\mathbb R)$ such that $x$ is not in $A$. Every Borel function $u$ is measurable with respect to $\delta_x$ and
$$
\int_\mathbb R u\,\mathrm d\delta_x=u(x).
$$
For example, if $u(x)=42$ for every $x\ne0$ and $u(0)=57$, then
$$
\int_\mathbb R u\,\mathrm d\delta_0=57.
$$
A: Probability density functions are applied by integrating over intervals to find your probability of being in the interval. Suppose you knew that you were precisely at the origin, with probability 100%? 
