The plane strain fundamental solution of instantaneous and a continuous point displacement discontinuity is presented in this paper. These solutions, together with the one of a fluid source, are obtained on the basis of a decomposition technique proposed by Biot, which separates the displacement field into a time independent part satisfying an elasticity equation, and an irrotational part governed by a diffusion equation. We begin the derivation by presenting a continuous edge dislocation. The continuous point displacement discontinuity is obtained by differentiating, along the direction of the cut, the corresponding edge dislocation solution. The instantaneous influence functions are determined by further differentiating with respect to time.