Hi,


Let $(f\_n)\_{n \geq 1}$ be a sequence of increasing functions defined on an interval, say $[0,1]$. 


Suppose that $\sum\_{n=1}^{\infty}f\_n(x)$ converges for all $x \in [0,1]$. Let $f:=\sum\_{n=1}^{\infty}f\_n$. 


It is well known that an increasing function defined on an interval is differentiable almost everywhere on that interval. But is it true that


$$f'(x)= \sum\_{n=1}^{\infty}f\_n'(x)$$
almost everywhere on $[0,1]$?


Any reference would help.


Thank you.



Differentiation of a series of increasing functions
Yes, see Theorem 4.1 on p. 177 of [this book](http://books.google.co.il/books?id=5ddbKSkaL8EC&pg=PR8&lpg=PR8&dq=%2522differentiating+series+of+monotone+functions%2522&source=bl&ots=_L6toIMGqu&sig=oTTRArrbAIjIKSFBRs6Egb819Ts&hl=iw&ei=IWYjTbK0BYGyhAfOnby3Dg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAYQ6AEwAA#v=onepage&q=%2522differentiating%2520series%2520of%2520monotone%2520functions%2522&f=false).


