Real Analysis, Folland Theorem 3.27 Properties of functions of Bounded Variation Background Information:
Taking $a = -\infty$ and considering the total variation as a function of $b$. To with $F:\mathbb{R}\rightarrow \mathbb{C}$ and $x\in\mathbb{R}$, we define $$T_F(x) = \sup\{\sum_{1}^{n}|F(x_j) - F(x_{j-1}|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\}$$ $T_F$ is called the total variation of $F$.
We observe that the sums in the definition of $T_F$ are made bigger if the additional subdivision points $x_j$ are added. Hence, if $a < b$, the deinition of $T_F(b)$ is unaffected if we assume that $a$ is always one of the subdivision points. It follows that $$T_F(b) - T_F(a) = \sup\{\sum_{1}^{n}|F(x_j) - F(x_{j-1}|:n\in\mathbb{N},a = x_0 < \ldots < x_n = b\}$$
Thus $T_F$ is an increasing function with values in $[0,\infty]$. If $T_F(\infty) = \lim_{x\rightarrow \infty}T_F(x)$ is finite, we say that $F$ is of bounded variation on $\mathbb{R}$, and we denote the space of all such $F$ by $BV$.
More generally, the supremum on the right side is called the total variation of $F$ on $[a,b]$. It depends only on the values of $F$ on $[a,b]$, so we may define $BV([a,b])$ to be the set of all functions on $[a,b]$ whose total variation on $[a,b]$ is finite.

Theorem 3.23 - Let $F:\mathbb{R}\rightarrow \mathbb{R}$ be increasing, and let $G(x) = F(x^+)$.
a.) The set of points at which $F$ is discontinuous is countable.
b.) $F$ and $G$ are differentiable a.e., and $F' = G'$.
3.25 Examples:
a.) If $F:\mathbb{R}\rightarrow \mathbb{R}$ is bounded and increasing, then $F\in BV$ (in fact, $T_F(x) = F(x) - F(-\infty)$).
b.) If $F,G\in BV$ and $a,b\in\mathbb{C}$, then $aF + bG\in BV$.
Lemma 3.26 - If $F\in BV$ is real-valued, then $T_F + F$ and $T_F - F$ are increasing.

Question:

Theorem 3.27
a.) If $F\in BV$ if and only if $Re F \in BV$ and $Im F \in BV$.
b.) If $F:\mathbb{R}\rightarrow \mathbb{R}$, then $F\in BV$ if and only if $F$ is the difference of two bounded increasing functions; for $F\in BV$ these functions may be taken to be $(\frac{1}{2}(T_F + F)$ and $\frac{1}{2}(T_F - F)$.
c.) If $F\in BV$, then $F(x^+) = \lim_{y\searrow x}F(y)$ and $F(x^-) = \lim_{y\nearrow x}F(y)$ exists for all $x\in\mathbb{R}$, as do $F(\pm\infty) = \lim_{y\rightarrow \pm\infty}F(y)$.
d.) If $F\in BV$, the set of points at which $F$ is discontinuous is countable.
e.) If $F\in BV$ and $G(x) = F(x^+)$, then $F'$ and $G'$ exist and are equal a.e.

Proof a.) I am sort of confused how I should show that $Re F \in BV$ and $Im F \in BV$ from just supposing $F\in BV$. I know that if $F\in BV$ then $\lim_{x\rightarrow \infty}T_F(x) = T_F(\infty) < +\infty$. But I am not sure how to add the real and imaginary parts of $F$ to show that they are also of bounded variation.
Proof b.) Suppose $F\in BV$ then $\lim_{x\rightarrow \infty}T_F(x) = T_F(\infty) < +\infty$ and from Example 3.25 we can write $$F = \frac{1}{2}(T_F + F) - \frac{1}{2}(T_F - F)$$
For the converse, I don't understand how if $F$ is taken to be the difference of these two increasing functions would should that $F\in BV$. What Folland does is refer back to the proof of Lemma 3.26 and states the inequalities for $x < y$ $$T_F(y) \pm F(y) \geq T_F(x) \pm F(x)$$ implies that $$|F(y) - F(x)| \leq T_F(y) - T_F(x) \leq T_F(\infty) - T_F(-\infty) < \infty$$ ok... so this shows that $F$ and $T_F\pm F$ is bounded. But how is Folland using the difference of $F$ to show that $F\in BV$?
For c,d,e Folland states that the result follows from a,b and Theorem 3.23. I can see how d and e may follow from Theorem 3.23 (not sure how to prove it though) but I don't see how the above results follow to prove c.
I am pretty lost with this theorem any suggestions is greatly appreciated.
 A: 
Theorem 3.27
a.) If $F\in BV$ if and only if $Re F \in BV$ and $Im F \in BV$.
b.) If $F:\mathbb{R}\rightarrow \mathbb{R}$, then $F\in BV$ if and only if $F$ is the difference of two bounded increasing functions; for $F\in BV$ these functions may be taken to be $(\frac{1}{2}(T_F + F)$ and $\frac{1}{2}(T_F - F)$.
c.) If $F\in BV$, then $F(x^+) = \lim_{y\searrow x}F(y)$ and $F(x^-) = \lim_{y\nearrow x}F(y)$ exist for all $x\in\mathbb{R}$, as do $F(\pm\infty) = \lim_{y\rightarrow \pm\infty}F(y)$.
d.) If $F\in BV$, the set of points at which $F$ is discontinuous is countable.
e.) If $F\in BV$ and $G(x) = F(x^+)$, then $F'$ and $G'$ exist and are equal a.e.

Proof:
a.) ($\Rightarrow$)
Suppose $F\in BV$. Then we have 
\begin{align*}
T_{ReF}(x) &= \sup\{\sum_{1}^{n}|ReF(x_j) - ReF(x_{j-1})|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\} \leq \\
& \leq \sup\{\sum_{1}^{n}|F(x_j) - F(x_{j-1})|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\} = \\
& = T_F(x)
\end{align*}
So, for all $x \in \mathbb{R}$, $ T_{ReF}(x) \leq  T_F(x)$ . So, $ \lim_{x\rightarrow \infty}T_{ReF}(x) \leq \lim_{x\rightarrow \infty}T_F(x)<+\infty$. So  $Re F \in BV$. 
In exactly the same way we prove that $Im F \in BV$.
($\Leftarrow$)
Now suppose $Re F \in BV$ and  $Im F \in BV$. Then, we have 
\begin{align*}
T_F(x) &= \sup\{\sum_{1}^{n}|F(x_j) - F(x_{j-1})|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\}  \leq \\
& \leq  \sup\{\sum_{1}^{n}(|ReF(x_j) - ReF(x_{j-1})|+ |ImF(x_j) - ImF(x_{j-1})|):n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\} \leq \\
& \leq  \sup\{\sum_{1}^{n}|ReF(x_j) - ReF(x_{j-1})|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\}+ \\
& \phantom{\leq\:}+ \sup\{\sum_{1}^{n}|ImF(x_j) - ImF(x_{j-1})|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\} = \\
& = T_{ReF}(x)+T_{ImF}(x)
\end{align*}
So, for all $x \in \mathbb{R}$, $T_F(x) \leq  T_{ReF}(x)+T_{ImF}(x)$.
So $\lim_{x\rightarrow \infty}T_F(x)\leq \lim_{x\rightarrow \infty}T_{ReF}(x) +  \lim_{x\rightarrow \infty}T_{ImF}(x)<+\infty$. So $F\in BV$.
b.) Let $F:\mathbb{R}\rightarrow \mathbb{R}$.
($\Rightarrow$)
Suppose  $F\in BV$. By lemma 3.26, $T_F + F$ and $T_F - F$ are increasing. So $\frac{1}{2} (T_F + F)$ and $\frac{1}{2} (T_F - F)$ are increasing and we have 
$$ F = \frac{1}{2} (T_F + F) -$\frac{1}{2} (T_F - F)$$
Now, to complete this part of the proof we must show that $\frac{1}{2} (T_F + F)$ and $\frac{1}{2} (T_F - F)$ are bounded. 
Since $F\in BV$, we have that $T_F$ is bounded, so all need is to prove that $F$ is a bounded function. This is a consequence of Lemma 3.26.
Note that, since  $T_F + F$ and $T_F - F$ are increasing functions, we have, for $y>x$, 
$$T_F(y) \pm F(y) \geq T_F(x) \pm F(x)$$
So 
$$T_F(y)- T_F(x) \geq \mp F(y)\pm F(x)$$
which means 
$$T_F(y)- T_F(x) \geq |F(y) - F(x)|$$
So we have 
$$|F(y) - F(x)| \leq T_F(y)- T_F(x) \leq  T_F(+\infty)- T_F(-\infty) < \infty$$
So $F$ is a bounded function.  
($\Leftarrow$) Suppose  $F$ is the difference of two bounded increasing functions. It is a direct consequence of 3.25 a.) and b.) that $F\in BV$.
Now, let us prove items c.) , d.) and e.) . 
Suppose $F\in BV$. Then by item a.),  $Re F \in BV$ and $Im F \in BV$. By item b.), we have that $Re F$ is  the difference of two bounded increasing functions $g_1$ and $g_2$ and also  $Im F$ is  the difference of two bounded increasing functions $g_3$ and $h_4$. So, we have that $F=(g_1-g_2)+i(g_3-g_4)$. 
Since $g_1$, $g_2$ , $g_3$ and $h_4$ are increasing functions, then, for $i=1,\ldots 4$, 
$$ g_i(x^+) = \lim_{y\searrow x}g_i(y) \textrm{ and } g_i(x^-) = \lim_{y\nearrow x}g_i(y) \quad \textrm{ exist and are finite, for all } x\in\mathbb{R}$$
And, since $g_1$, $g_2$ , $g_3$ and $h_4$ are bounded increasing functions, then, for $i=1,\ldots 4$,
$$g_i(\pm\infty) = \lim_{y\rightarrow \pm\infty}g_i(y)<\infty$$
Since $F=(g_1-g_2)+i(g_3-g_4)$, we have that $F(x^+) = \lim_{y\searrow x}F(y)$ and $F(x^-) = \lim_{y\nearrow x}F(y)$ exist and are finite, for all $x\in\mathbb{R}$, as do $F(\pm\infty) = \lim_{y\rightarrow \pm\infty}F(y)<\infty$.
So we have proved c.)
Now, by 3.23 a.) we have that, for  $i=1,\ldots 4$, the set of points at which $g_i$ is discontinuous is countable. Since $F=(g_1-g_2)+i(g_3-g_4)$, we have that 
the set of points at which $F$ is discontinuous is countable. So we have proved d.)
Now, define $G(x) = F(x^+)$ and, for  $i=1,\ldots 4$, define $h_i(x)=g_i(x^+)$. 
Then
$$G(x) = F(x^+)=(g_1(x^+)-g_2(x^+))+i(g_3(x^+)-g_4(x^+))= (h_1(x)-h_2(x))+i(h_3(x)-h_4(x))$$
Now, by 3.23 b.) we have that, for  $i=1,\ldots 4$,  $g_i'$ and $h_i'$ exist and are equal a.e.. So, since  $F=(g_1-g_2)+i(g_3-g_4)$ and $G= (h_1-h_2)+i(h_3-h_4)$, we have that $F'$ and $G'$ exist and are equal a.e.
So we have proved e.). 
A: For (a), use the fact that for any complex number $z$, we have the inequality $\max\{|\text{Re}z|,|\text{Im}z|\} \leq |z| \leq |\text{Re}z| + |\text{Im}z|$.
For the converse direction of (b), write $F = I - D$ where $I$ and $D$ are bounded and increasing. Then if $x_0 < x_1 < \ldots x_n$, we have
$$\begin{aligned}
\sum_{k=1}^{n}|F(x_k)-F(x_{k-1})|& = \sum_{k=1}^{n}|I(x_k) - D(x_k) - I(x_{k-1}) + D(x_{k-1})|\\
&\leq \sum_{k=1}^{n}|I(x_k) - I(x_{k-1})| + \sum_{k=1}^{n}|D(x_k) - D(x_{k-1})| \\
&= \sum_{k=1}^{n}(I(x_k) - I(x_{k-1})) + \sum_{k=1}^{n}(D(x_k) - D(x_{k-1}))\\
&= (I(x_n) - I(x_0)) + (D(x_n) - D(x_0)) \\
&\leq \lim_{x \to \infty}I(x) - \lim_{x \to -\infty}I(x) + \lim_{x \to \infty}D(x) - \lim_{x \to -\infty}D(x)
\end{aligned}$$
The limits on the RHS exist and are finite since $I$ and $D$ are increasing and bounded.
For (c), note that a step discontinuity is the only kind of discontinuity possible in an increasing function; in particular this means that the left-hand and right-hand limits exist at all points. By (b), the same is true of any real-valued BV function, and therefore by (a), the same is true of the real and imaginary components of $F$.
For (d), note that by (a), the real and imaginary parts of $F$ have bounded variation. As these are real-valued, example 3.25(a) shows that each is the difference of two increasing functions. Now $F$ cannot be discontinuous at a point $x$ unless one of these four increasing functions is discontinuous at $x$. Since each increasing function is continuous except possibly on a countable set, the same is true of $F$, because the union of finitely many countable sets is countable.
For (e), again decompose $F$ into real and imaginary components, and then decompose each component into the difference of two increasing functions, then do the same for $G$, then apply 3.23(b) to each of the four pairs of increasing functions.
