I am considering the following question for a while but fruitless:

Suppose a function $f:C\subseteq\Re^2\to\Re$ is the pointwise limit of continuous functions, where $C$ is compact. That is to say, there is a sequence of continuous function $\{f_n\}$ such that $f_n\to f$ pointwise on $C$. I was wondering what conditions on $f$ (except that $f$ is continuous) can be imposed so that

(i) we can find a uniformly bounded sequence of continuous functions $\{g_n\}$ such that $g_n\to f$ pointwise; or

(ii) we can find a monotone sequence of continuous functions $\{h_n\}$ (i.e., $h_n(x)\ge h_{n+1}(x)$ for all $x\in C$) such that $h_n\to f$ pointwise?

I hope someone can give me some hint or reference on this. Many thanks!


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