Is there a continuous version of the Uniform distribution? The Uniform distribution is not differentiable. Is there a differentiable distribution that approximates a Uniform?
 A: The standard continuous uniform distribution $\text{U}(a,b)$ distribution has a continuous CDF that is differentiable (in the regular sense) at all points except the edges of its support, $x = a, b$.  Since probability theory defines density functions using Radon-Nikodym derivatives, we can still ascribe values to the density function even at these end-points.
In view of the use of Radon-Nikosym derivatives in probability, I cannot think of any context where this lack of (regular) differentiability of the CDF would really matter.  Nevertheless, if you really want to approximate the uniform distribution with a distribution having a fully differentiable distribution function (in the regular sense), you could approximate the density with a mixture distribution (e.g., a mixture of evenly spaced normal distributions).
A: This is a familiar problem in theoretical mathematics, where it helps the analysis when you don't have to worry about lack of differentiability.  The standard solution, sometimes called "mollification," is to convolve the density with a scaled, zero-centered, infinitely differentiable density (often of compact support). By setting the scale close to zero you can make the approximation as close as you like.

The figure is a sequence of graphs of mollified Uniform$(0,1)$ density functions using a Gaussian mollifier with standard deviations $1/4$ (green), $1/10$ (gold), $1/25$ (red), and $0$ (blue: the original Uniform PDF).
It is easy to show (use integration by parts in the formula for a convolution) that when the mollifier is infinitely differentiable everywhere (aka "smooth"), so is the mollified function.
The existence of such families of mollifiers means that for most purposes you don't really have to consider non-differentiable densities (or even singular distributions, which by definition do not have a density everywhere) when thinking about properties of distributions.  Singular distributions might indeed be "edge cases" but they can be approached as limits of distributions with smooth densities.

This method is particularly congenial in statistical applications because many properties of the mollified distribution are easily computed.  As an example, since the variance of the mollifier is proportional to the square of its scale, if we pick a standard mollifier with unit variance (as in this example), the variance of the mollified Uniform distribution equals the variance of the Uniform distribution (here, $1/12$) plus the square of the scale.  Thus, you know immediately that mollification with a Gaussian of standard deviation $1/25$ (as in the red curve) will add only $1/(25)^2$ to the variance of the Uniform distribution.  You can select the standard deviation to be so small that the change in variance it induces is negligible for your purposes.
