$\int_{-\infty}^\infty f(x) dx = 1$ and $f(x) \ge 0$ for all $x$.

$\int_{-\infty}^\infty f(x) dx = 0$ and $f(x)$ is continuous.

$f(x) \le 1$ for all $x$ and $\int_{-\infty}^\infty f(x) dx = 1$.

$\int_{-\infty}^\infty f(x) dx = 1$ and $\lim_{x \to \infty} f(x) = 0$.