A continuous random variable has the CDF:
$\hspace{10mm} F(x) = \left\{\begin{array}{rcl} 0 & & x < 0 \\ cx^2 & &0 \leq x \leq 2 \\ 1 & & x > 2 \end{array}\right\}$
Find the value of the positive constant $c$.
For the function to be continuous I understand that we will evaluate 4c = 1 and get the value of our constant.
My question is: What if there are two such arguments between 0 and 1. If we have two constants "c" mentioned in the same cumulative distribution function, is it impossible to find the value of the constant?
Note: the constants are the same. Both are c.
Say you had something like
$$F(x)=\begin{cases} 0 & x < 0 \\ c_1 x^2 & 0 \leq x<0.5 \\ c_2 x & 0.5 \leq x<1 \\ 1 & x \geq 1 \end{cases}.$$
Say you are again given that the CDF is continuous. Then you can get an equation for continuity at $0.5$ and another equation for continuity at $1$. These equations read $c_1/4=c_2/2,c_2=1$. Thus $c_1=2,c_2=1$.