Surjective Proof Homework: Show f is Surjective on (c,d)

[analysis001]

Homework Statement

Suppose f: (a,b)→R where (a,b)$\subset$R is an open interval and f is a differentiable function. Assume that f'(x)≠0 for all x$\in$(a,b). Show that there is an open interval (c,d)$\subset$R such that f[(a,b)]=(c,d), i.e. f is surjective on (c,d).

Homework Equations

f is surjective if for all y$\in$R there exists an x$\in$X such that f(x)=y.

The Attempt at a Solution

I think I'm supposed to use ε and δ for this proof but I'm not sure where to start. Any clues would be great! Thanks.

 

[Dick]

Homework Statement

Suppose f: (a,b)→R where (a,b)$\subset$R is an open interval and f is a differentiable function. Assume that f'(x)≠0 for all x$\in$(a,b). Show that there is an open interval (c,d)$\subset$R such that f[(a,b)]=(c,d), i.e. f is surjective on (c,d).

Homework Equations

f is surjective if for all y$\in$R there exists an x$\in$X such that f(x)=y.

The Attempt at a Solution

I think I'm supposed to use ε and δ for this proof but I'm not sure where to start. Any clues would be great! Thanks.

I don't think you need ε and δ. Start by thinking about continuous functions (like f, since it's differentiable) and the Intermediate Value Theorem. The Mean Value theorem will come in handy too.