I am working with the probability density function that depends on non-deterministic ($v$) and random ($x$) parameters (and random parameters depend on non-det parameters):
$Pr(v)=\int_{G(v)} dP(v)$,
where $G (v)$ is the "goal" region, the probability of getting into it must be determined and that, in turn, depends on $v$ and $x$.
So in general case we have:
$Pr(v)=\int_{g(v)}^{m(v)} f(v,x)dx,$
where $f(v,x)$ is a probability density function.
In my previous question - Using the Leibniz integral rule to prove smoothness of the function. I was trying to prove that $Pr(v)$ is a smooth function, however I need to consider an assumption that :
$\forall v \, \int_{-\infty}^{\infty} f(v,x)dx<\infty \implies \forall v \, \int_{-\infty}^{\infty} \frac{d}{dt}f(t,x)|_{t=v}dx<\infty,$
For example: the above implication holds for exponential distribution function $f(v,x)$, because it converges to zero very quickly ($\lambda e^{−\lambda x}$).
MY QUESTION: Can somebody help me to find a counterexample of a probability density function, where this implication does not hold?
I will really appreciate your help!