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What is the correct definition of bounded probability density function:
$\sup_{x} f(x)<\infty$. If this is the correct definition of bounded probability density function, can you give the example of a "well-known" probability density function which is not bounded?
An unbounded probability distribution has support on all real numbers. In this case, he Normal distribution would be unbounded, for instance. Instead, the uniform distribution would be bounded.
Here is an example of an unbounded density function from the Gamma distribution:
$$f(x) = \text{Ga}(x|\tfrac{1}{2},1) = \frac{1}{\sqrt{\pi x}} \cdot \exp(-x) \quad \quad \quad \quad \quad \text{for } x > 0.$$
For this density function we have $\lim_{x \downarrow 0} f(x) \rightarrow \infty$ so $\sup_x f(x) = \infty$.
