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Math Help - creating a jump discontinuity

  1. #1
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    creating a jump discontinuity

    for the question: State a function that has an infinite discontinuity at x = 0 and a jump discontinuity at x = 3

    I believe that creating an infinite discontinuity at x=0 requires x in the denominator of a rational function. Do I create a jump discontinuity by creating a piece-wise function?
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  2. #2
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    Re: creating a jump discontinuity

    Hey kingsolomonsgrave.

    You will want to create a definition for the function at x < 3 and one at x = 3 and possibly at x > 3 if you want a point discontinuity as opposed to just a normal one.

    You can use what is called the Heaviside function to do this:

    Heaviside step function - Wikipedia, the free encyclopedia

    It basically does what a piece-wise function does by transferring this information into H(x-a) where H is the Heaviside function.
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  3. #3
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    Re: creating a jump discontinuity

    Thanks! Can I use the Heaviside step function to create a function whose graph has BOTH an infinite discontinuity at x=0 AND a jump discontinuity at x=3? (or infinite discontinuity at x=c and jump continuity at x=k where c is not = to k and both are constants)
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    Re: creating a jump discontinuity

    Yes you can do it: Your function will be f(x)*[1 - h(x-k)] + g(x)*h(x-k) where an example for f(x) having an infinite discontinuity at x = c would 1/(x-c) and g(x) could be for example g(x) = - 1/(x-c) which will contain a noticable "jump". There are many different choices for f(x) and g(x) and this pair is just one example.
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