What should be the formal definition of continuous uniform distribution pdf value at upper bound? What is the most formal (and coherent with probability theory) definition for the value of $f(b)$ where $b$ is the upper bound of the support of the continuos uniform distribution $\mathcal U(a,b) ?$
We can choose:
\begin{align}f(b)& = 1/(b-a) ~~\text{or}\\
 &= 0~~~~~~~~~~~~~~~~\text{or}\\
 &= \text{undefined}.\end{align}
 A: Partially answered in comments:

No pdf is defined, as a function, at any point.  As a convention, a pdf is often represented as a function that is continuous wherever possible.  Because no pdf for this distribution can be continuous at either endpoint, it cannot be uniquely defined at either endpoint according to such a convention.  In this sense the last bullet is rigorously the answer, but the other two bullets are perfectly valid, too.


The PDF is defined as the Radon-Nikodym derivative of the probability measure with respect to Lebesgue measure.  A more elementary characterization of the PDF is that if there exists a function $f$ for which the distribution function $F(x) = \int_{-\infty}^x f(x)\mathrm{d}x,$ then $f$ is a PDF for $F.$ Even when you use elementary Riemann integration you may arbitrarily modify any such $f$ at a finite number of points without changing that defining relationship with $F.$ Your reference concerns only the restricted set of absolutely continuous distributions, btw.


One conclusion is that it is worthwhile understanding the nuances of how a PDF represents a density.

– whuber
