Although the question actually popped up in a course about evolution equations, it seemed most natural to ask this in the context of joint distributions. Namely:
Suppose $X$ and $Y$ are two variables with joint distribution $f_{X,Y}(x,y)$, which is jointly analytic in $x$ and $y$. Does this imply that the marginal distributions $f_X(x)$ and $f_Y(y)$ are continous?
I could create counterexamples if the statement is strengthened to imply differentiability (see below), or starting from smooth joint distributions, but the latter all had essential singularities (take for example this, as provided by hcl14 ).
However, the derivatives of $f$ in $x$ and $y$ would not necessarily be integrable, so an obvious expansion in power series would not work, unfortunately. For (a little convoluted) example:
Consider \begin{equation} f_{X,Y}(x,y):=\frac{2 \sin (x y)e^{-y} }{x^2+1}, \qquad x\geq0, y\geq 0 \end{equation} Then $f$ is analytic in $x$ and $y$ (however it is not holomorphic, it has poles in $x=\pm i$), and a straightforward calculation verifies that it defines a joint distribution. However, its power series in $y$ around $0$ is not integrable in $x$: \begin{equation} f_{X,Y}(x,y)=\frac{2 x y}{x^2+1}-\frac{2 x y^2}{x^2+1}+\mathcal{O}[y]^3 \end{equation} Finally, the marginal distribution $f_Y(y)$ (with a little help from Mathematica) seems not be differentiable in $0$.
Any insight is greatly appreciated!