Implications of differentiable demand function on the utility function properties

Suppose you know that the (Marshallian) demand function $x(p,m)$ that satisfies the consumer's problem of utility maximization is such that $\frac{\partial x_i(p,m)}{\partial m}$ is always well-defined for each good $i\in\{1,\dots,L\}$. In other words, the demand function is differentiable with respect to income.

What does this necessarily implies about the properties of the utility function $U(x_1,\dots,X_L)$?

• For simplicity, we can restrict the analysis to the case of non-negative goods, that is $x_i(p,m)\in R^+, \forall p_i\geq 0, m\geq 0, \forall i\in\{1,\dots,L\}$ – GabMac Oct 3 '17 at 9:17