We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditionsψ(x1, 0)ψ†(x2, t)±,T. We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special casex1= 0, we express correlation functions with Neumann boundary conditionsψ(0, 0)ψ†(x2, t)+,T, in terms of solutions of nonlinear partial differential equations which were introduced in [1] as a generalization of the nonlinear Schrödinger equations. We generalize the Fredholm minor determinant formulae of ground state correlation functionsψ(x1)ψ†(x2)±,0in [2], to the Fredholm determinant formulae for the time and temperature dependent correlation functionsψ(x1, 0)ψ†(x2, t)±,T,t ∈R,T ≥0.