Convergence in real analysis If I have the sequences of functions $\{f_j\}\in C^{\infty}(\mathbb{R})$ and $\{g_j\}\in C^{\infty}(\mathbb{R})$. Assuming also that $f,g\in C^0(\mathbb{R})$.
Under the assumption that $f_j\rightarrow f$ and $g_j\rightarrow g$ on every bounded interval $[a,b]$ as $j\rightarrow \infty$ and $f'_j(x)=g_j(x)$, then I want to show that $f'(x)=g(x)$.
Thanks in advance
 A: Set $ u_n (x) = u (x, n) $,  where $ n \in \mathbb {N} $
You are essentially asking whether you could differentiate the limit  $\lim_{n \to \infty} u_n(x)$ and what is it equal to.
This might help :
This is a stronger version (uniformly integrability condition) of what you need:
If a sequence of  absolutely continuous functions {$f_n$} converges pointwise to some $f$  and if the sequence of  derivatives {$f_n’$}  converges almost everywhere to some  $g$ and if {$f_n’$} is uniformly integrable then $\lim\limits_{n\mapsto \infty} f_n’ = g= f’$ almost everywhere. Where the derivative of  $f$ is $f’$. If the convergence is pointwise and $ g $ is continuous then $ f'$ = $ g $ everywhere.
Proof : by FTC  $f_n(x) – f_n(a) = \int_a^x f_n’ dx$
By Vitali convergence theorem : $\lim\limits_{n\mapsto \infty}\int_a^x f_n’ dx = \int_a^x g dx$
Therefore $\lim\limits_{n\mapsto \infty}( f_n(x) – f_n(a))= \int_a^x g dx$
$f(x)-f(a) = \int_a^x g dx$
$f(x)’=g$ almost everywhere
If the convergence is pointwise and $ g $ is continuous then $ f'$ = $ g $ everywhere.
A: If the convergence is uniform then write $f_j(x)=f_j(a)+\int_a^{x} f_j'(t)\, dt$ and take limits. You get $f(x)=f(a)+\int_a^{x} g(t)\, dt$. Since $g$ is continuous (by uniform convergence) we see that $f$ is differentiable and $f'=g$. 
A: Thank you both, It make sense.
If I now want to expand the problem to $f_j, g_j$ and $f,g$ to being locally integrable functions, i.e. they are in $L_{loc}^1(\mathbb{R})$. 
Under the assumption that $f'_j(x)=g_j(x)$ in the sense of distributions, I wanna show that $f'(x)=g(x)$ in $\mathbb{R}$ in the sense of distributions.
I have assumed that $f_j\rightarrow f$ in $L_{loc}^1(\mathbb{R})$ as $j\rightarrow \infty$ if for every bounded interval interval $[a,b]$ we have that $\int_{a}^b |f_j(x)-f(x)|dx\rightarrow 0$.
Thanks in advance.
