Change of variables with a continuous but not differentiable mapping

Question

Suppose that Y is a continuous random variable with a density function $f_{Y}(y)$. We transform $Y$ by the following mapping

\begin{equation} Y^{*} = \left \{ \begin{array}{ll} \alpha Y + \beta & \text{ if } Y < y^{0} \\ Y & \text{ if } Y > y^{0} \end{array} \right . \end{equation} where $\alpha$ and $\beta$ are known constants such that the mapping is continuous, but not differentiable at $Y = y^{0}$. How do I find the density function of $Y^{*}$?

My concern is how the non-differentiable point $Y=y^0$ affects the resulting density of $Y^∗$. My solution for the density of $Y^∗$ is the following:

$f_{Y^{*}}(y^{*}) = f_{Y}(y^{*}) I\{ Y^{*} > y^{0} \} + \frac{1}{\alpha}f_{Y}\left ( \frac{y^{*} - \beta}{\alpha} \right) I\{ Y^{*} < \alpha y^{0} + \beta \}. $

Answer 1

The transform leads to a cumulative distribution function that is non-differentiable in the point where you split the transformation.

The density function will have a jump discontinuity in that point. Possibly there are ways to still have the density defined in a way (e.g. use the directional continuity of the cumulative distribution function or an average as you did) but the discontinuity remains.

Below you see an example when we perform such transformation on the normal distribution