A common step in solving many PDEs is to write a multivariable solution function as a product of two more or single variable functions. For example, if given,
$$ \alpha\,\frac{\partial U(x,t)}{\partial x} = \frac{\partial U(x,t)}{\partial t}, $$
one may begin be writing,
$$ U(x,t) = X(x) T(t). $$
My question is, what are the necessary conditions for writing a multi variable function as a product of several single variable functions? Furthermore, are there any cases where this cannot be done?