Let $g \colon \mathbb{R}_+ \to \mathbb{R}_+$ be a real concave and strictly increasing function on $\mathbb{R}_+$.
Let $ \mathbb{R}^E_+$ be the set of all real-valued nonnegative functions on a topological space $E$.
If there is a sequence $\{ f_n \} \subset \mathbb{R}^E_+$ such that their composition functions $\{g \circ f_n\}$ converges to a limiting composite function $ g \circ f \in \mathbb{R}^E_+$ pointwise (i.e., $\forall x \in E$, $| g (f_n (x)) - g(f(x))| \to 0$ as $n \to \infty$) , then could it imply $\{f_n \}$ converges in $f$ pointwise (i.e., $| f_n(x) - f(x)| \to \infty$ as $n \to \infty$)? If not, then when it could imply?
How about the case when a sequence $\{ f_n \} \subset \mathbb{R}^E_+$ such that their composition functions $\{g \circ f_n\}$ converges uniformly to a limiting function $ g \circ f \in \mathbb{R}^E_+$ on the set $ E$? Does it imply that $\{f_n \}$ converges uniformly to $f$ on $E$? If not, when it could imply?
The answer is yes. For then $g$ will be invertible and continuous with same properties for $g^{-1}.$