Which two conditions must a function meet to be considered a valid probability density function (pdf)?

The function must be positive for some values, and its maximum value must be $1$

The function must be non-negative everywhere, and its integral over all real numbers must be $1$

The function must be non-negative everywhere, and its maximum value must be less than $1$

The function must be continuous everywhere, and its integral over all real numbers must be $0$