\begin{definition}[Definition:Probability Density Function]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X: \Omega \to \R$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $\Omega_X = \Img X$, the image of $X$.
Then the '''probability density function''' of $X$ is the mapping $f_X: \R \to \R_{\ge 0}$ defined as:
:$\forall x \in \R: \map {f_X} x = \begin {cases}
\ds \lim_{\epsilon \mathop \to 0^+} \frac {\map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2} } \epsilon & : x \in \Omega_X \\
0 & : x \notin \Omega_X
\end {cases}$
\end{definition}