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Math Help - Continuous Functions

  1. #1
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    Continuous Functions

    I'm having some trouble with this question:

    Let f : [a,b] \rightarrow \mathbb{R} be\ a\ continuous\ function\ such\ that\ f(x)>0 for\ all\ x \in [a,b]. Show\ that\ there\ is\ a\ positive constant\ c\ such\ that\ f(x) \geq c for\ all\ x \in [a,b]. Give\ examples\ to\ show\ that\ the\ conclusion\ fails if\ [a,b]\ is\ replaced\ by\ (0,1] or [0,\inf ).
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  2. #2
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    This is just the extreme value theorem. Since f is continuous and positive on a compact set [a,b], it achieves a positive minimum in [a,b].

    An example of a function for which it fails is f(x) = \sqrt{x} on (0,1] or f(x) = {1\over 1+ x} on [0,+\infty).
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  3. #3
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    Quote Originally Posted by acevipa View Post
    I'm having some trouble with this question:

    Let f : [a,b] \rightarrow \mathbb{R} be\ a\ continuous\ function\ such\ that\ f(x)>0 for\ all\ x \in [a,b]. Show\ that\ there\ is\ a\ positive constant\ c\ such\ that\ f(x) \geq c for\ all\ x \in [a,b]. Give\ examples\ to\ show\ that\ the\ conclusion\ fails if\ [a,b]\ is\ replaced\ by\ (0,1] or [0,\inf ).
    How you go about proving this much depends on what you already know about compact intervalls like [a;b], and continuous functions.
    For example, there is a theorem that says that for a continuous function on a compact interval [a;b], there are x_{\text{min}},x_{\text{max}}\in[a;b], such that f(x_{\text{min}})\leq f(x)\leq f(x_{\text{max}}), for all x\in [a;b].
    So, if you can use that theorem, you can just put c := f(x_{\text{min}}), but if you cannot use that theorem, you have to (essentially) prove it.
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