Expert: Socrates - 5/27/2005

Question
I just have a couple of questions that I need help with.

1.  I would like to know how would I prove that
   f(x) = x^2 + 1 is a continuous function

2.  If I had something like...
    Let f: Z -> Z, where Z is the set of
    integers and f(x) = x^3    
    Is f(x) a one-to-one function?  If so, then
    how?

Answer
1) This is easy if you know that the sum and product of continuous functions is continuous. Let h(x)= 1 ,g(x)= x.  Assuming you know that g and h are continuous ,then observe that f(x) = g(x)g(x) + h(x). g(x)g(x) is continuous because a product of continuous functions is continuous. f(x) = g(x)g(x) + h(x) is then continuous because it is a sum of continuous functions.
An epsilon , delta argument is also possible , but more complicated.

2)Yes, f is one to one. Suppose f(x) = f(y). Then
x^3 = y^3
x^3 - y^3 = 0
(x-y)(x^2 + xy + y^2) = 0
so either x - y = 0 or x^2 + xy + y^2 = 0
The quadratic formula shows that x^2 + xy + y^2 = 0 has no real solutions, so that leaves us with x - y = 0 , and then x = y , which proves that f is one to one.