distribution function

[captain_epsilon]

ok so i'm working in futility on a problem that is making zero sense to me in a chapter section entitled Expectations of Discontinuous Functions and Mixed Probability Distributions of this book Mathematical Statistics With Applications 6th edition. The problem is #127 and i can't even get through part a.

F(y) = 0 if y < 0
y^2+.1 if 0 <= y < .5
y if .5 <= y < 1
1 if y >= 1

Find the discrete and continuous distribution function components of F(y). I struggled for a long time to understand the way that the solution was derived but could not. Wow this problem was depressing. By what reasoning has the author deduced the coefficients of the discrete and continuous components?? Note that I don't need to be convinced that the given coefficients work but rather how he got them.