Shannon Entropy Continuity Constraint I have the following problem: I want to find the probability density $p$ which maximizes the Shannon entropy 
\begin{equation}
S := - \int_{x_b}^{x_c} dx ~ p(x) \log (p(x))
\end{equation}
under the following constraints:


*

*normalization

*$p(x_b) = v$ for some fixed value $v$

*$p$ is continuous.


Usually, such problems can be solved using a Lagrange multiplier. 
My problem is: how can I impose the continuity condition in terms of Lagrange multipliers? 
 A: The optimization problem, as stated, is ill posed. Specifically, one can get a sequence of continuous functions that satisfy your pointwise constraints and get higher and higher entropy, while converging to a non-continuous function. To see this, take functions that are constant everywhere, except in an epsilon-sized ball around the measurement locations $x_b$, where they continuously transform to the measurement values, then back to the constant. These continuous functions are converting to the constant function, except for point discontinuities at the $x_b$ points.
This is reflective of a larger issue of entropy on functions, which is that entropy only cares about the values of a function, but not where those values are located in the domain spatially. If you chop up a function and move the pieces around, the entropy will remain the same. 
In general, a good idea in this case is to regularize the problem with a regularization that penalizes the variation or nonsmoothness in the function. For example, Laplacian regularization will penalize how much the value of a function at a point deviates from the local average of the function, favoring functions that are smoother over functions that are less smooth. 
One minimizes the modified objective function,
$$
-S(p) + \frac{\alpha}{2}||Rp||^2 \\
$$
where $R$ is the regularization operator like a power of the Laplacian, and with the same constraints.
