I found that the initial condition $u(x,0)=\phi(x)$ need to satisfy atmost finite number of discontinuities for the existence and uniqueness of exact solution through Fourier transforms for the heat equation $$u_t(x,t)=Du_{xx}(x,t),~(x,t) \in \mathbb R \times \mathbb R^+.$$
But I am not getting the exact reason for these requirements. Is it for the integrability of the convolution integral?