More help on Elementary Analysis?

Question

See my other question also, please.

15.a. Show that f = max(f,0) + min(f,0). This particular way of writing f is fairly useful; the functions max(f,0) and min(f,0) are called the positive and negative parts of f.
b. A function f is nonnegative if f(x) is > or = 0 for all x. Prove that any function can be written f = g - h, where g and h are nonnegative, in infinitely many ways.

23. Suppose that f(g(x)) = I, where I(x) = x. Prove that
a. if x isnt equal to y, then g(x) isnt equal to g(y)
b. every number b can be written b = f(a) for some number a

Answer 1

15.a. for a given x, suppose f(x)>=0. Then max(f(x),0)=f(x), and min(f(x),0)=0, so max(f(x),0)+min(f(x),0) = f(x). Similar for f(x)<0.
b. One way is to set g=max(f,0) and h=-min(f,0)=max(-f,0). Infinitely many other ways can be found by a simple modification of this g and h, that preserves their nonnegativity.

23.a. suppose x is not equal to y, but g(x)=g(y). Then f(g(x))=f(g(y)), but this will lead to a contradiction.
b. f(g(b))=b, so let a=g(b).