How to decompose a CDF into discrete and continuous parts? I understand that any C.D.F may be represented in the form 
$$F(x) = p_1F^d(x) + p_2F^c(x)\,,$$ 
where $F^d(x)$ represents discrete c.d.f , $F^c(x)$ represents continuous c.d.f and $p_1+ p_2=1$.
What is the procedure to decompose the c.d.f in such a form, assuming that it is already a mixed c.d.f?
Consider this cdf :
$$
F(x)=
\begin{cases}
0 &,\text{ if }x<0 \\
x^2+0.2 &,\text{ if }0\le x<0.5 \\
x &,\text{ if }0.5\le x<1 \\
1 &,\text{ if }x\ge 1
\end{cases}
$$
How can we express this cdf in the form mentioned above? 
 A: Note your CDF has two jumps: One of size $\frac{1}{5}$ at $x = 0$ and another of size $\frac{1}{20}$ at $x = \frac{1}{2}$. If we subtract off the jumps at these points we are left with a continuous function,
$$F_c(x) = \begin{cases}0 & x < 0 \\ x^2 & 0 \leq x < \frac{1}{2} \\ x - \frac{1}{4} & \frac{1}{2} \leq x < 1 \\ \frac{3}{4} & x \geq 1 \end{cases} $$ 
You can think of this is as scaled down continuous CDF (but not a valid one since it does not approach $1$ for $x\rightarrow\infty$). What is the scaling factor, $p_2$ here? In particular, $F_c(x) = p_2*F^c(x)$, where $F^c(x)$ is a valid continuous CDF. 
Now consider the jumps we subtracted off earlier. We can create a right-continuous step function using these jumps, 
$$F_d(x) = \begin{cases}0 & x < 0 \\ \frac{1}{5} & 0 \leq x < \frac{1}{2} \\ \frac{1}{4} & x \geq \frac{1}{2}\end{cases} $$
You can think of this is as scaled down discrete CDF (but not a valid one since it does not approach $1$ for $x\rightarrow\infty$). What is the scaling factor, $p_1$ here? In particular, $F_d(x) = p_1*F^d(x)$, where $F^d(x)$ is a valid discrete CDF. 
Putting things together, you obtain your desired decomposition, $F(x) = p_1*F^d(x) + p_2*F^c(x)$. 
