Integrability of Derivative of a Continuous Function Let $f$ be continuous on $[a,b]$ and has finite derivative a.e. on $[a,b]$.
Let $f_n(x)=n[f(x+1/n)-f(x)] $ s.t. $f_n$ be uniformly integrable on $[a,b]$.
I want to show : $f'$ is Lebesgue integrable.
(I noted that $f_n\rightarrow f'$ pointwise.)
 A: This answer refers to a previous version of the (now edited) question, without assuming uniform integrability of $(f_n)$.
This is in general not true. A counterexample is $f(x) = x \sin \frac1x$ on $[0,1]$ (with $f(0)=0$). Then $f'(x) = \sin \frac1x - \frac1x \cos \frac1x$ is not Lebesgue-integrable over $[0,1]$, because $\int_0^1 |f'(x)| \, dx = \infty$ in this case.
You can even modify this example to be differentiable everywhere, using $g(x) = x^2 \sin \frac{1}{x^2}$. Then $g'(x) = 2x \sin \frac1{x^2} -\frac2x \cos\frac1{x^2}$ and again $\int_0^1 |g'(x)|\, dx = \infty$.
A: By the Mean Value Theorem, for each positive integer $n$ and for each $x \in [a,b]$, 
there exists $z_{n,x} \in (x, x+1) $ so that 
$$ 
f_n(x)= f'(z_{n,x}).
$$
It follows that $|f_n(x)| \leq M$ for all $n$ and for all $x\in [a,b]$.
Since $\{f_n\}$ converges pointwise to $f'$ on $[a,b]$, the Bounded Convergence Theorem yields
$$ 
\int_a^b f' = \lim_{n \to \infty } \int_a^b f_n
$$
which shows that $f'$ is Lebesgue integrable.
