In Friedman's book PDE’s of Parabolic Type, the Green function $G$ for the heat equation is said to satisfy \begin{equation*} G_t - \Delta G = 0 \end{equation*} in $\Omega\times(0,T)$ and $G=0$ on $\partial\Omega\times(0,T)$, where $\Omega$ is a open bounded subset of $\mathbb{R}^n$ with smooth boundary.
Furthermore, it is stated that for a continuous function $f$ with compact support, i.e., $f\in C_c(\Omega)$ we have \begin{equation} \lim_{t\rightarrow0} \int_\Omega G(x,y,t) f(y) \, \mathrm{d}y = f(x) \end{equation} for every $x\in \Omega$.
Question 1. Shouldn't it be enough for $f$ to be in $C(\Omega)\cap L^\infty(\Omega)$ in order for the last property above to be true?
Question 2 In papers they often state that for $f\in L^1(\Omega)$ \begin{equation} \lim_{t\rightarrow0} \int_\Omega G(x,y,t) f(y) \, \mathrm{d}y = f(x) \end{equation} for a.e. $x\in \Omega$. How does one show this?