Setup
Let $f(y)$ be three times differentiable with bounded third derivative in a neighborhood of $y \in \mathbb R$.
At this time we want to find examples of $f$ under special conditions. That condition is when either $f$ or $f'$ (or both) is not bounded. We also impose the condition that $f$ is a probability density function.
If either $f$ or $f'$ (or both) are bounded, examples abound, e.g., normal and exponential distributions.
Question
Could you please give me some hints or specific examples of finding such $f$ examples? Please let me know if there is anything I am missing.
$f(x)=x^3$ is an example where $f(x)$ and $f'(x) = 3x^2$ are not bounded, but $f'''(x)=6$ is. However, it does not satisfy the strict constraint; p.d.f.