Proving f-g is Continuous & f Inverse Existence

[pivoxa15]

Homework Statement

a) If f and g are continuous functions how do you show f-g is continous?

b) if f is continuous then f inverse exists?

The Attempt at a Solution

a) Do you look at the sets that f and g maps points in the domain into?

b) I think so.

 

[HallsofIvy]

(a) How about looking at |f(x)- g(x)- (f(a)- g(a))|\le |f(x)- f(a)|+ |g(x)-g(a)| as just about every calculus book does? Since f is continuous (at a) you can make that first absolute value less than any \epsilon, since g is continuous (at a), you can make the second absolute value less than any epsilon.<br /> <br /> (b) What about y= x² with domain all real numbers? Is it continous? Does it have an inverse? What do the DEFINITIONS of &quot;continous&quot; and &quot;inverse function&quot; have to do with each other?

 

[pivoxa15]

For y=x^2 it has an inverse for [0,infinity) but not all the real numbers, so no for b)