Does the Pontryagin principle still hold when the function inside the objective integral is discontinuous?

Question

For example:

I have a control problem: $$ max \int_0^1 (U(y, \theta) - t)f(\theta) d\theta$$ s.t. $$t'= -U_1(y, \theta, b)u $$ and $$y' = u$$

Now, t and y are state variables and u is the control variable.

If the PDF $f(\theta)$ is discontinuous in $\theta$, but y, t, and u are still continuos on $\theta$, does the pontryagin principle still holds? If not, how to solve this control problem? Thanks!