Distribution to density to expected

A random variable X has the cumulative distribution function

for

for

for

Calculate the variance of X.

The answer specifies that the density function is

if x=1

if 1<x<2

otherwise

Then

I got the f(x) = x-1 part, and I got how to calculate the variance after you have the expected values, but I'm lost on other questions.

My questions are:

**Where do we get if x=1?** F(1) = 0.5, but I can't figure out why f(1) would equal 0.5.

**What is the rule for putting parts of the stepwise density function into the expected value equations?** I don't know what the rule is called so I don't know how to review it. We're adding the slope at a single point to the slope over a big area, which is something I can't quite work out visually.