All functions are locally $s$-harmonic up to a small error
  We show that we can approximate every function $f\in C^{k}(\bar{B_1})$ with a $s$-harmonic function in $B_1$ that vanishes outside a compact set.   That is, $s$-harmonic functions are dense in $C^{k}_{\rm{loc}}$. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature. 
Serena Dipierro, Ovidiu Savin and Enrico Valdinoci
math.AP