Let $f_n$ be a sequence of positive bounded functions and $g(x)$ be some positive bounded functions.
Suppose, we have an identity \begin{align} \limsup_{n \to \infty} \, ( f_n(x)-f_n(0))= g(x)-g(0) \end{align} for all $x$.
Now I want to take derivative of both sides with respect to $x$. Clearly, we have that \begin{align} \frac{d}{dx}\limsup_{n \to \infty} \, ( f_n(x)-f_n(0))= \frac{d}{dx}g(x) \end{align}
My question: Is it true that \begin{align} \frac{d}{dx}\limsup_{n \to \infty} \, f_n(x)= \frac{d}{dx}g(x) \end{align} For example, this is true if $f_n(0)$ is a constant for all $n$. Is it true in general. In other, words I am asking if \begin{align} \frac{d}{dx}\limsup_{n \to \infty} \, ( f_n(x)-f_n(0))=\frac{d}{dx}\limsup_{n \to \infty} \, f_n(x) \end{align}
My opinion is that no. Since, limsup's are subadditive and not additive that is, in general, we only have \begin{align} \limsup_n (a_n+b_n) \le \limsup_n a_n +\limsup_n b_n. \end{align}