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Discontinuity to continuity

  1. Nov 14, 2004 #1
    just a basic question, so if i'm asked to find 2 functions that are discontinus, but when added together, becomes continuous, how do I approach that?

    can I say like, let

    F(x) = 1 for x =< 0, and f(x) = 0 for x > 1.
    G(x) = 1 for x =<0 and g(x) = 0 for x = 1.

    can I just somehow "add" f + g and say that is continuous? I dont know....tips?
  2. jcsd
  3. Nov 15, 2004 #2

    matt grime

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    Let f be your favourite discontinuous function, then let g=1-f.

    Prove g is discontinuous, and hence find a continuous function that is the sum of two continuous ones.

    Your f and g, did you mean to capitalize them? Note g doesn't have the same domain as f; g isn't defined for any positive real numbers, and hence neither is f+g.
  4. Nov 15, 2004 #3


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    Well, you can't just say it's continuous- because it isn't!

    (f+ g(x)= 2 for x<0, 1 if x= 0 and 0 if x>0
    (I've switched the last "1" to "0". The f and g you give are not defined between 0 and 1. Unless that's a typo, I have the uncomfortable feeling that you don't know what is meant by "defining" a function.

    Taking f(x)= 1 for x<= 0, f(x)= 0 for x> 0, which is not continuous at x= 0,
    try matt grimes' suggestion. What is g(x)= 1- f(x)?
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