Definition 5.3.2. Probability "Density" Function.
Given a continuous random variable \(X\) on a space \(R\text{,}\) a probability density function on \(X\) is given by a function \(f:R \rightarrow \mathbb{R}\) such that:
\begin{align*}
& \forall x \in R , f(x) \gt 0\\
& \int_{R} f(x) dx = 1\\
& A \subset R \Rightarrow P(X \in A) = \int_{A} f(x) dx
\end{align*}
For \(x \not\in R\text{,}\) you can use the convention \(f(x)=0\text{.}\)