So you meant the space of Lipschitz continuous functions is dense in that of uniformly continuous functions w.r.t. the supremum norm. Is the space of uniformly continuous functions dense in that of continuous functions w.r.t. the supremum norm? Thank you so much for your elaboration! Please see my question here?

I'm very interested in this quote, but unable to understand. Could you please explain it?

IHi, are $x_1,\ldots,x_m$ in the expression $g(x):=f(x)-px_1\dots x_m$ constant, or are they dependent on $x$? it seems to me that $x_1,\ldots,x_m \in \mathbb R$.

Could you please explain how to go from $g(x):=f(x)-px_1\dots x_m$ to $\delta_{t_me_m}\dots \delta_{t_1e_1}g(x) = \delta_{t_me_m}\dots \delta_{t_1e_1}f(x)-p\,t_1\dots t_m$?

After long time digesting your proof using finite difference operator, I have combined it with my previous attempt to to give my it a try. I have posted my proof here. If you don't mind, please have a look at it. Thank you so much! By the way, I'm just exposed to Real Analysis, so your proof is quite advanced for me.

Thank you @DanieleTampieri for your encourage! Honestly, I proved the theorem in case mixed partial derivatives are of order $2$ here.