Suppose $F$ is an increasing function on $[a, b]$. (a) Prove that we can write $$ F=F_{A}+F_{C}+F_{J} $$ where each of the functions $F_{A}, F_{C}$, and $F_{J}$ is increasing and:
(i) $F_{A}$ is absolutely continuous.
(ii) $F_{C}$ is continuous, but $F_{C}^{\prime}(x)=0$ for a.e. $x$.
(iii) $F_{j}$ is a jump function.
If we let $$ j_{n}(x)=\left\{\begin{array}{cl} 0 & \text { if } x<x_{n} \\ \theta_{n} & \text { if } x=x_{n} \\ 1 & \text { if } x>x_{n} \end{array}\right. $$ then we define the jump function associated to $F$ by $$ J_{F}(x)=\sum_{n=1}^{\infty} \alpha_{n} j_{n}(x) . $$
I am kind of sure that I have to use the property that an increasing function can have only countably many jump discontinuity. Given an increasing order of discontinuities $\{x_n\}$ of $F$ i.e., $x_n\le x_{n+1}$ and $f$ has discontinuities at $x_n$ then we can represent the function $$ F(x)=f_n(x); x_n\le x\le x_{n+1} $$ and $f_n$ is continuous on $(x_n,x_{n+1})$.
Now we can define a function $$G(x)=\left\{\begin{array}{cl} g_1(x):=f_1(x) & \text { if } a=x_1\leq x<x_{2} \\ g_{n}(x) & \text { if } x_n\le x\le x_{n+1} \end{array}\right. $$ where $g_n(x)$ is a function defined on $(x_n,x_{n+1})$ having the same slope of $f_n(x)$ on $(x_n,x_{n+1})$ so that $G(x)$ is continuous basically I am removing the jump discontinuities of $F$. So we can write $F(x)=G(x)+J(x)$ where $$J(x)=f_n(\frac{x_n+ x_{n+1}}{2})-g_n(\frac{x_n+ x_{n+1}}{2}); x_n\le x\le x_{n+1} $$
then $J(x)$ is a step function. I am wondering if I can at all transform this thing into the solution I want. Let me know if there is any other way. Please help!
