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Simple analysis continuity problem

  1. Oct 5, 2009 #1
    1. The problem statement, all variables and given/known data
    If f is a real function which is continuous at a element R and if f(a)<M for some M element of R, prove that there is an open interval I containing a such that f(x)<M for all x element of I.


    2. Relevant equations
    Extreme value theorem, intermediate value theorem, definition of continuity


    3. The attempt at a solution
    I have no idea how to solve this.
     
  2. jcsd
  3. Oct 5, 2009 #2
    Draw out the scenario. You have a continuous function with some point a on the x-axis, f(a) on the y-axis. Draw a horizontal line through the point f(a). We can think of M as a horizontal line somewhere above f(a). Let D be the distance between f(a) and M. If we can make f(x) within D of f(a), then M will be a bound for f(x) as well. But this is guaranteed by the continuity of f (epsilon-delta definition).
     
  4. Oct 5, 2009 #3
    Thanks for the explanation. The problem is clearer now to me. What I still do not get is how the epsilon delta definition guarantees it. ?? Sorry if I can't follow through... I'm new to this.
     
  5. Oct 6, 2009 #4
    Oh I think I got it. Thanks
     
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