Let $X\sim Uniform[0,1]$ and $z:[0,1]\rightarrow[0,1]$ be a non-monotonic function (but with very nice features such as continuity, differentiability, etc).
If a function $\alpha$ is defined to be $$\alpha(t)=P[z(X)\leq t],$$ Is there anyway that we can get a closed form representaiton of $\alpha'(t)$?
Yes, there is:
$$\alpha'(t) = \mathcal I(t \in [0, 1]) \sum_{x_i: z(x_i) = t} \left| \frac{dz}{dt}(x_i) \right|^{-1}$$
Here, $\mathcal I$ is the indicator function that takes value $1$ if $t$ is within the range $[0, 1]$ and the sum is over all values of $x_i$ such that $z(x_i) = t$.
The way one arrives at this answer - if you look at Wikipedia's change of variables page, there's an equation for the density of random variables under non-monotonic transformations. Essentially, we add up density contributions from all $x$ that can map to $t$ and given the piecewise monotonicity, we can apply the usual Jacobian adjustment on each monotonic segment of $z$.