In Evans PDE book, Section 2.3, Theorem 2. It says that if $f\in C^2_1(\mathbb{R}^n\times [0,\infty))$ and $f$ has compact support, then $$ u(x, t)=\int_{0}^{t} \int_{\mathbb{R}^{n}} \Phi(x-y, t-s) f(y, s) d y d s $$ is the solution of $u_t-\Delta u=f$. Here $\Phi$ is the fundamental solution.
My question is, can we relax the assumption on $f$?
For example,if $f$ is discontinuous like $f=0$ on $x<0$ and $f=1$ on $x>0$, does this formula still hold?