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Continuety of splitted function

  1. Dec 29, 2008 #1
  2. jcsd
  3. Dec 29, 2008 #2


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    The English for "splitted function" is "piecewise defined" function although here it is more "pointwise" defined:
    f(x)= 0 if x is irrational, 1/n if x is ration, x= m/n reduced to lowest terms (and n is assumed to be positive).

    As I said in a previous response, [itex]lim_{x\rightarrow a} f(x)= L[/itex] if and only if [itex]lim_{n\rightarrow\infty} f(a_n)= L[/itex] for any sequence [itex]{a_n}[/itex] converging to x. Obviously for any sequence, [itex]a_n[/itex]., of irrational numbers converging any x, that limit is 0.

    So in order that this be continuous, the limit as we approach along rational numbers must also be 0 and the function value must be 0.

    Here's a hint. If x= m/n, given any [itex]\epsilon> 0[/itex] there are only a finite number of possible N such that [itex]N< 1/epsilon[/itex] (so [itex]1/N> 1/\epsilon[/itex]) and for each such N there are only a finite number of M such that M/N is within [itex]\delta[/itex] of m/n. Use that to prove that the limit always exists and is always equal to 0.
  4. Dec 30, 2008 #3
    i know the definition of bound and of continuity
    i cant understand what the last hint means and how to use it
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