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Thread: Intervals and Continuous Functions

  1. #1
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    Intervals and Continuous Functions

    Hello,

    I'm trying to prove that if $\displaystyle f(x)$ is continuous and if $\displaystyle f(a)=c,f(b)=d$ then $\displaystyle f \colon [a,b] \longrightarrow [c,d]$ or $\displaystyle f:]a,b[ \longrightarrow ]c,d[$ in brief the image of a closed interval is a closed interval if f is continuous and the same with an open interval. Since I'm doing this just for fun I'm wondering if the definition of a continuous function is enough to prove it?( The definition I'd use is : f is continuous at x if and only if $\displaystyle \forall \varepsilon \exists \delat>0, |f(x_1)-f(x_2)| \leq \varepsilon, \forall x_1,x_2 \in ]x- \delta,x + \delta[ \cap \mathbf{D}$ (with $\displaystyle f \colon \mathbf{D}\subset \mathbf{R} \longrightarrow \mathbf{R} $.
    If you can give me the full proof I'll gladly look at it !

    Thanks
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  2. #2
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    Are you sure this is true?
    What about $\displaystyle f(x)=sinx$ with domain $\displaystyle (- 2 \pi, 2 \pi)$? What about constant functions?
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  3. #3
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    hoooo so I think I need f bijective then. Well thanks now I think I know what to do.
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  4. #4
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    Quote Originally Posted by sunmalus View Post
    hoooo so I think I need f bijective then. Well thanks now I think I know what to do.
    It is a well known theorem that the continuous image of a connected set is connected. The non-trivial connected sets in the real numbers,$\displaystyle \mathbb{R}^1$, are simply the non-degenerate intervals.

    That does what you are trying to do.
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