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Math Help - Prove a connected set is an interval

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    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?
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  3. #3
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    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.
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  4. #4
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    Anyone has the solution to this one?
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