If I have a function $g_{\rho} \in C_c^{\infty}(\mathbb{R^n})$ $($i.e $C^{\infty}$ function compactly supported in $\mathbb{R^n}$$)$.
Defined by: $g_{\rho}:=\frac{1}{\rho^n}g(\frac{x}{\rho})$, where $g \in C_c^{\infty}(\mathbb{R^n})$, and $\int_{\mathbb{R^n}}g(x)dx=1$.
And if I defined $f_{\rho}:=g_{\rho}*f$, where $f \in C(\Omega)$ $($or $C(\overline{\Omega}))$, $\Omega \subset \mathbb{R^n}$.
Then what can we say about the limit of $f_{\rho}$ as $\rho \rightarrow 0$? And in what sense?
My Dr. said: $f_{\rho} \rightarrow f$ as $\rho \rightarrow 0$ locally uniformly in $\Omega$. And then continued to say: this means the pointwise convergence on all $\Omega$
But how might this be true?