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I am looking for an example of the following: Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the pushforward density and $p$ is continuous as well. What if continuity is strengthened to differentiability?

I am looking for an example of the following: Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the pushforward density and $p$ is continuous as well. What if continuity is strengthened to differentiability?

Edit: Intuitively this seems impossible, just by continuity considerations; e.g. pick a neighborhood where $q_1$ and $q_2$ differ, and invoke bijectivity of $f$.

I am looking for an example of the following: Given a probability density function $p$, find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=f_*q_2$, where $f_*$ is the pushforward density.

I am looking for an example of the following: Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the pushforward density and $p$ is continuous as well. What if continuity is strengthened to differentiability?

Given a probability density function $p$, give an example of a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=f_*q_2$, where $f_*$ is the pushforward density.

I am looking for an example of the following: Given a probability density function $p$, find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=f_*q_2$, where $f_*$ is the pushforward density.