Let $g(z)$ be a continuous function on $\mathbb R^n\setminus \{0\}$. $$ \int_{B_R(0)} |g(z)| dz \leq C_1 $$ for some constant $C_1$, and with $B_R (0)$ being the ball of radius $R$ centered at the origin. Outside the ball
$$|g(z)|\leq C_2 | z|^k ,\; \forall z\in \mathbb{R}^n \setminus B_R (0) .$$
Does it follow that $$ \lim_{\epsilon \rightarrow 0}\int_{\mathbb{R}^n} g(\epsilon z+y) \exp(-\pi | z |^2) dz = g(y)\,? $$
If yes can you please give me a reference for the proof? Or if not give a counterexample.
Without further assumptions on $g$ this is false. Take $g(x) = 0 $ for $x\neq 0$ and $g(0)=1$. Then $g$ is continuous in $\mathbb R^n \setminus \{0\}$ (in fact it is smooth there) but $g=0$ a.e. so $$\int_{\mathbb R^n} g(\varepsilon z) e^{-\pi \vert z\vert^2} \, dz = 0 \neq g(0) . $$