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Implied continuity

  1. Nov 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Let f be a real valued function whose domain is a subset of R. Show that if, for every sequence xn in domain(f) \ {x0} that converges to x0, we have lim f(xn) = f(x0) then f is continuous at x0.


    2. Relevant equations
    Book definition of continuity:
    "...f is CONTINUOUS at x0 in domain(f) if, for every sequence xn in domain(f) converging to x0, we have limnf(xn)=f(x0)..."


    3. The attempt at a solution
    The statement lim f(xn) = f(x0) would suggest that f(x0) exists, so leave that part of continuity aside for now.

    What's the trick to get from domain(f) \ {x0} to domain(f) to satisfy the defintion?
     
  2. jcsd
  3. Nov 2, 2009 #2
    Use the limit hypothesis to help show the standard epsilon-delta definition of continuity is satisfied at x0. You will use the epsilon-delta definition of the limit.
     
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