"relative frequency distribution" of function values Let $\ f(x):U\longrightarrow\mathbb{R}$ be a continuous real-valued function over a closed interval $U\subseteq\mathbb{R}$.
I would like to define a "relative frequency distribution" function, $\ \mathcal{F}:\mathbb{R}\longrightarrow[0,1]$ which measures in some sense "how often" the function $f$ takes a value $y$.
The idea is made more precise as follows.
Let $\{J_k\}$ be a family of disjoint intervals that covers the reals:
\begin{equation}
\bigcup_{k\in\mathbb{Z}}J_k=\mathbb{R}\qquad J_i\ \cap J_j = \emptyset\quad\text{if}\quad i\neq j
\end{equation}
Let also $y_k\in J_k$ be a value in each interval.
Define
\begin{equation}
F(y_k):=\frac{\lambda[f^\leftarrow(J_k)]}{\lambda[U]}
\end{equation}
where $\lambda$ is the standard (Lebesgue) measure.
Finally, I'd like to "define", with a lot of handwaving,
\begin{equation}
\mathcal{F}(y) :=\lim_{\lambda[J_k]\rightarrow 0}F(y_k)
\end{equation}
Of course, this definition makes no sense because $y$ is not well defined (how to pick $y$ as the interval $J_k$ becomes smaller?) and also because it all collapses to zero...but I hope that the sense of it is clear.
My question is: how could I formally define the function $\mathcal{F}$? The discrete version of the idea works because it's all about counting how many points lie in the preimage of each $y$, but I don't know how to extend it to the continuous case, or even if it's possible.
 A: The concept that best describes what you want is the pushforward measure.
More precisely, you want the usual Lebesgue (length) measure $\mu$ pushed forward by the function $f$.
The pushforward is denoted $f_*\mu$ and defined so that $f_*\mu(A)=\mu(f^{-1}(A))$.
That is, $f_*\mu(A)$ is the size of the "level set" $f^{-1}(A)$ and therefore describes how often $f$ takes values in $A$.
The benefit of this approach is that the pushforward measure exists in great generality.
Whether a function like the one you defined exists is a harder question.
In general it doesn't.
For example, if $f$ is constant, what would $F$ be?
Since you are working over the real line, you can use functions more comfortably.
If $\mu(U)<\infty$, you can define $G(x)=f_*\mu((-\infty,x])$, which is the amount of times values at most $x$ are obtained.
This behaves much like a probability distribution.
If $G$ happens to be differentiable (which is by no means guaranteed), then perhaps $G'$ might have properties suitable to you.
I can't give much more details since I don't know what you want to use the tool for.
