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Analysis Question

  • Thread starter Math_Geek
  • Start date
23
0
[SOLVED] Analysis Question

1. Homework Statement
Let f:Reals to Reals be a continuous at x=a, and further suppose f(a)>0. Show there exists an interval I about x=a such that f(x)>0 for all x in I.

2. Homework Equations

none

3. The Attempt at a Solution
i know the defn of continuity but I am not sure how to show this interval exists.
I need HELP!
 

Answers and Replies

1,631
4
well, this is quite easy to show, just use the definition of the limit, and chose epsylon in such a manner that it will allow you to do this. i.e [tex]0<\epsilon<f(a)-0[/tex].

[tex]\forall\epsilon>0, \ \ also \ \ for 0<\epsilon<f(a),\exists\delta(\epsilon)>0[/tex] such that

[tex]|f(x)-f(a)|<\epsilon, \ \ whenever \ \ 0<|x-a|<\delta[/tex] so from here we have:

[tex] -\epsilon<f(x)-f(a)<\epsilon =>f(a)-\epsilon<f(x)<f(a)+\epsilon[/tex] but look now how we chose our epsilon, [tex]\epsilon<f(a)=>f(a)-\epsilon>0[/tex] so, eventually

[tex]0<f(a)-\epsilon<f(x), \ \ whenever, \ \ \ xE(a-\delta,a+\delta[/tex].

hope this helps.
 
Last edited:
23
0
wow thanks
 
1,631
4
23
0
how do I know it is in the interval around a?
 
1,631
4
how do I know it is in the interval around a?
But notice that x is from the interval [tex] (a-\delta,a+\delta)[/tex] which actually includes a. MOreover, it works only for that interval around a, for other intervals we are not sure.
 
Last edited:

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