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Existence of limits and continuity

  1. Aug 14, 2012 #1
    1. The problem statement, all variables and given/known data
    #1. If limit[x->a]f(x) exists, but limit[x->a]g(x) doesnt, limit[x->a](f(x)+g(x)) doesn't exist. T/F? (Proof or example please)

    #2. prove that if f is continuous, then so is |f|

    #3. f(x) = [[x]]+[[-x]] for what a does limit[x->a]f(x) exist? Where is f discontinuous?

    2. Relevant equations

    3. The attempt at a solution
    #3 confuses me the most; my first thought is the function = 0 for all x so the limit should exist for all a and it should be 0, which means it should be continuous everywhere, but there is a theorem that states if f(x) is disc. on (a,b) and g(x) is disc. on (a, b) then f(x) + g(x) is disc. on (a,b) and since they are both disc. on integer values so should be their sum...so which thought is correct and why?
  2. jcsd
  3. Aug 14, 2012 #2


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    What if f(x) = 0 if x rational and 1 if x irrational
    and g(x) = 1 if x rational and 0 if x irrational?
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