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Simple Question about Continuity

  1. Apr 6, 2006 #1
    I'm just working through some differentiability questions and have a quick question - are functions continuous at a cusp or corner? I know that functions are not differentiable at cusps or corners because you cannot draw a unique tangent at these points, but I'm not sure about continuity. From my understanding, a function is continuous if at point a if

    1. The limit as x approaches a exists
    2. f(a) exists or is defined
    3. the limit of x approaching a = f(a)

    Can you define the function at a cusp or corner and thus have a continuous function?

    Thanks!
    Rozy
     
  2. jcsd
  3. Apr 6, 2006 #2

    0rthodontist

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    Yes, you can.
     
  4. Apr 6, 2006 #3

    arildno

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    On a side-note, you can construct functions that are everywhere continuous, but nowhere differentiable..
     
  5. Apr 6, 2006 #4
    Thanks

    Thanks for responding! :smile:
     
  6. Apr 6, 2006 #5

    HallsofIvy

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    For example f(x)= |x| has a "cusp" or corner at x= 0. Of course, f(0)= |0|= 0 so the function is certainly define there- and continuous for all x.
     
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