The problem is
A random variable X has the cdf $F(x)=\frac{x^2-2x+2}{2}\quad\text{if}\quad1\leq x<2$
and $F(x)=0$ when $x<1$, $F(x)=1$ when $x\geq 2$.
Calculate the variance of X(the answer is $\frac{5}{36}$)
My question is
What is the relationship between with the cdf and pdf when cdf has a jump at the point x=1;
What is the definition of $E[X^2]$ when pdf is not continuous at the point x=1?
I think this is a case for the r.v. is partially discrete and partially continuous, so, I don't know the definition. Maybe it needs some knowledge of lebesgue stieltjes integral, I'm not sure. Could you please help me? Thank you so much!