An integro-differential equation without continuous solutions (Q727713)

The authors construct a nonsymmetric kernel so that the solution of the integro-differential equation given below has a solution that is discontinuous at the origin: \[ \int_{ \mathbb{R}^n} (u(x+y)-u(x)-y. \bigtriangledown u(x) \chi _{B_1}(y)) ~K(x,y)~dy=f(x) \] in \(B_1\). The required kernel is constructed as a sum of three functions and satisfies the following property \[ \frac{ \lambda}{|y|^{n+ \alpha} } \leq K(x,y) \leq \frac{ \wedge}{|y|^{n+ \alpha}} \] for some \(\alpha \in (0,1)\).