# Surjective Proof

1. Apr 27, 2014

### analysis001

1. The problem statement, all variables and given/known data
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).

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

3. 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.

2. Apr 27, 2014

### Dick

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.

Last edited: Apr 27, 2014