Question

Prove that if f is a continuous function on an interval, then so is |f|.

Answer 1

Let f:E→ℝf:E→R a continuous function and II and interval, I⊆EI⊆E. Then f(I)f(I) is also an interval.

I'm not sure if I've understood completely what I have to prove. So, I need to prove that f takes all the values in f(I)f(I), which is easy, using the intermediate value theorem.

But how do I know that ∀x∈f(I)∀x∈f(I), then f(x)∈If(x)∈I ? Isn't it necesary to also prove this? I know for sure that ∀λ∈f(I)∀λ∈f(I) , there is an xλxλ such that f(xλ)=λf(xλ)=λ, given the fact that I=[a,b]I=[a,b] , and f(I)=[f(a),f(b)]f(I)=[f(a),f(b)]. That is, forbsure, every value in f(I)f(I) is taken by f. So, what I don't know how to prove is how do I know that for any value in II , the function sends me for sure in f(I)f(I)??