I've learned probability in the past but I did not learn these functions in the analysis fashion i.e., specify domain and range of your function. So I've been confused about it ever since, especially when it comes to generalizing it to ALL possible PDFs.
I know that the PDF $f$ is associated with a continuous random variable, say $X$. The domain of $f$ is the range of $X$, where the range is a "continuous set" (maybe we can require it to be a connected set; not too sure about the topological characteristic on the range either).
$$f: \text{range}(X) \to \text{something}$$
From the few examples I've seen, I believe that $\text{something}$ is always a real number, no matter how large the dimension of the range becomes $i.e., \text{range}(X) = \mathbb{R}^n$, at least for the functions I've seen (Gaussian, Uniform, Weibull PDF).
Is this true and why? Could it be a subset of $\mathbb{R}$? Any reference that explicitly talks about this helps!
I view it like this: Random variable is defined on probability space, takes values in $\mathbb R^n$ (for real-valued random variable).
Since $X$ is measurable, $P(X\in A)$ is defined for measurable subset $A$ of $\mathbb R^n$. This induces a measure on $\mu_X$ on $\mathbb R^n$. So $P(X\in A) = \int_A \mu_X（dx)$. This measure is the distribution of X.
IF that measure is absolutely continuous with respect to the lebesgue measure then there is a a radon nikodym derivative $f(x)$ such that $\int_A \mu_X(dx) = \int_A f(x) dx$.
$f(x)$ is a member of an equivalnce class so might be hard to define explicitly. But domain is all of $\mathbb R^n$. Might be zero on large parts of this domain though. Think of $\int f(x) dx $ as a probability measure, so has to have all properties of a probability measure.
Most importantly, $\int_{\mathbb A} f(x) dx = P(X\in A) \in [0,1] $ for all $A \subset \mathbb R^n$ (measurable), so $f(x)$ must be non-negative a.e.
So, I'd say domain is all of $\mathbb R^n$, and $f(x) \in [0,\infty)$ almost everywhere.