Bounded function with finitely many discontinuities is integrable $\overset{?}{\Rightarrow}$ density of continuous distribution function is not unique


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?