The density function of the distribution function of a continuous random variable is not uniquely defined.
A new density function can be obtained by changing the value of the function at finite number of points to some non-negative value, without changing the integral of the function. We then get a new density function for the same continuous distribution.
Does this follow from the theorem-
A bounded function with finite number of discontinuities over an interval is Riemann integrable.
or is there a different theorem supporting the above claim? Is the theorem a sufficient justification?