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Rigorous definition of continuity on an open vs closed interval

  1. Dec 1, 2011 #1
    Let I be an open interval and f : I → ℝ is a function. How do you define "f is continuous on I" ?

    would the following be sufficient? :

    f is continuous on the open interval I=(a,b) if [itex]\stackrel{lim}{x\rightarrow}c[/itex] [itex]\frac{f(x)-f(c)}{x-c}[/itex] exists [itex]\forall[/itex] c[itex]\in[/itex] (a, b)

    is this correct?

    Also, what about the case of a closed interval I? In that case, can you just add to the above statement that:

    [itex]\stackrel{lim}{x\rightarrow}a^{+}[/itex] f(x) = f(a)
    and
    [itex]\stackrel{lim}{x\rightarrow}b^{-}[/itex] f(x) = f(b)

    ???
     
  2. jcsd
  3. Dec 1, 2011 #2

    dextercioby

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    The definition you gave is for differentiability on an open interval. The definition for continuity is with delta and epsilon. The [ itex ] for lim looks like [itex] \displaystyle{\lim_{x\rightarrow a}} [/itex].
     
  4. Dec 1, 2011 #3
    But differentiability implies continuity?
    Anyway, then would this be right:

    f is continuous on the open interval I=(a,b) if |[itex]\frac{f(x)-f(c)}{x-c}[/itex] - f'(c)|< ε when |x-c|<δ [itex]\forall[/itex] c[itex]\in[/itex] (a, b)


    and what about my use of the right and left hand limits for the case of a closed interval [a,b]? would that be correct?
     
  5. Dec 1, 2011 #4

    dextercioby

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    I was referring to the http://en.wikipedia.org/wiki/Continuous_function Weierstrass epsilon-delta definition for continuity.
     
  6. Dec 1, 2011 #5
    ohhhh nvm. it can be continuous on (a,b) but not differentiable, like the abs value of x.
    Ok then it would just be:
    f is continuous on (a,b) if [itex] \displaystyle{\lim_{x\rightarrow c}} [/itex] f(x) = f(c) [itex]\forall[/itex] c[itex]\in[/itex] (a, b)

    right???
     
  7. Dec 1, 2011 #6

    dextercioby

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    Yes, but for that limit to make sense, you have to use the topology, that is compare values of f(x_1) and f(x_2) when x_1 and x_2 get arbitrarily close to each other. That's what the epsilon-delta definition does.
     
  8. Dec 1, 2011 #7
    we didnt learn about topology yet so idk what is f(x_1) and the like...
     
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