- #1
NihalRi
- 134
- 12
Homework Statement
We've been given a set of hints to solve the problem below and I'm stuck on one of them
Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0
Hint
let set S = {x∈[a,b]:f(x)≤0}
let c = supS, show c∈[a,b]
Homework Equations
∀ε>0, ∃δ>0 such that ∀x∈(a,b), if |x-p| <δ then |f(x)-f(p)|<ε
The Attempt at a Solution
I have not gotten very far
want to show a<c<b
c≥x ∈[a,b]: f(x)≤0
a<x<b
want to show a<x≤c<b, so show c<b because as sup it's already bigger than x
f(a) ≤f(x)≤ 0 < f(b) ⇒ f(x)<f(b)
I can see it is true intuitively but I'm struggling with showing this properly. I would appreciate your help greatly.