Integrate across the origin e.g. from -2 to 2.

Integrate[f[x] D[D[HeavisideTheta[x], x], x], {x, -2, 2}]
(* Out[5]= -Derivative[1][f][0] *)

Integrating that particular expression with 0 as an endpoint of the integration interval is in effect GIGO ("garbage in, garbage out").