# Homework Help: Continuous functions are borel

1. May 20, 2012

### stukbv

1. The problem statement, all variables and given/known data
Take f: (a,b) --> R , continuous for all x0in (a,b)
and take (Ω = (a,b) , F = ( (a,b) $\bigcap$ B(R)) where B(R) is the borel sigma algebra
Then prove f is a borel function

3. The attempt at a solution

I know that continuity of f means that for all x in (a,b) and all ε>0 there exists a δ>0 such that |x-x0| < δ implies |f(x)-f(x0| < ε

And I want to show that {x in (a,b) s.t f(x) < c } is in F

But Then I am stuck, how would I use these facts to help me ?

Thanks in advance for any help

2. May 21, 2012

### micromass

What do you know about continuous functions?? Do you know that for a continuous function and G open that $f^{-1}(G)$ is open??