# Thread: Prove a connected set is an interval

1. ## Prove a connected set is an interval

Problem: Shown a connected subset in R is an interval.

Proof so far:

I let a subset A to be connected in R, then A has Intermedian Value Property, thus for every continuous function f:A->A, f(A) is an interval.

Now since f(A) is an interval, A is an interval...

Am I right?

2. Originally Posted by tttcomrader
Problem: Shown a connected subset in R is an interval. Proof so far: I let a subset A to be connected in R, then A has Intermedian Value Property, thus for every continuous function f:A->A, f(A) is an interval. Now since f(A) is an interval, A is an interval...
Any interval in R is one of eight types: (-oo,a], (-oo,a), (a,b), (a,b], [a,b), [a,b], [b,oo), or (b,oo). That is, there are four types of bounded intervals of reals and four types of unbounded intervals of reals. How does what you wrote prove that if A is a connected set in R the A is one of those types?

3. Since f(A) is an interval, that means the image is of [a,b].

Now f(A) = {f(x) : x in A}, does that means the domain is of an interval in the form of [a,b]?

I'm sorry, I'm really lost at this section. The test is coming up this thursday, and I simply have too many questions that I do not know how to answer.

The problems I'm posting up now are practice problems from the book btw.

4. Anyone has the solution to this one?