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Thread: Continuous Functions

  1. #1
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    Continuous Functions

    I'm having some trouble with this question:

    $\displaystyle Let f : [a,b] \rightarrow \mathbb{R} $ $\displaystyle be\ a\ continuous\ function\ such\ that\ f(x)>0$ $\displaystyle for\ all\ x \in [a,b].$ $\displaystyle Show\ that\ there\ is\ a\ positive$ $\displaystyle constant\ c\ such\ that\ f(x) \geq c$ $\displaystyle for\ all\ x \in [a,b].$ $\displaystyle Give\ examples\ to\ show\ that\ the\ conclusion\ fails$ $\displaystyle if\ [a,b]\ is\ replaced\ by\ (0,1] or [0,\inf ).$
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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 $\displaystyle f(x) = \sqrt{x}$ on $\displaystyle (0,1]$ or $\displaystyle f(x) = {1\over 1+ x}$ on $\displaystyle [0,+\infty)$.
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  3. #3
    Super Member Failure's Avatar
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    Quote Originally Posted by acevipa View Post
    I'm having some trouble with this question:

    $\displaystyle Let f : [a,b] \rightarrow \mathbb{R} $ $\displaystyle be\ a\ continuous\ function\ such\ that\ f(x)>0$ $\displaystyle for\ all\ x \in [a,b].$ $\displaystyle Show\ that\ there\ is\ a\ positive$ $\displaystyle constant\ c\ such\ that\ f(x) \geq c$ $\displaystyle for\ all\ x \in [a,b].$ $\displaystyle Give\ examples\ to\ show\ that\ the\ conclusion\ fails$ $\displaystyle if\ [a,b]\ is\ replaced\ by\ (0,1] or [0,\inf ).$
    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 $\displaystyle x_{\text{min}},x_{\text{max}}\in[a;b]$, such that $\displaystyle f(x_{\text{min}})\leq f(x)\leq f(x_{\text{max}})$, for all $\displaystyle x\in [a;b]$.
    So, if you can use that theorem, you can just put $\displaystyle c := f(x_{\text{min}})$, but if you cannot use that theorem, you have to (essentially) prove it.
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