Continuous functions are borel

[stukbv]

Homework Statement

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

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

 

[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??