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    Derivative of a distribution

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    Let f(x) and g(x) differentiable functions on R. Evaluate the derivative, in distribution sense, of the function h(x):

    h(x) = f(x) if x > 0
    h(x) = g(x) if x â?¤ 0.

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    Solution Preview

    Let's denote the action of a distribution T on a test function u as <T,u>. Then the derivative of T, denoted as DT, is defined by declaring that it acts on test functions as:

    <DT, u> = - <T, Du>

    This definition makes it consistent wih the partial integration formula when T and DT are ordinary functions (note that, by definition, test functions tend to zero at infinity fast enough that you don't get boundary terms when doing partial integration).

    In this problem the distribution is the function h which is, in general, discontinuous at x = 0. if u is a general test function, we have by definition:

    <Dh,u> = - <h, Du>

    Since on the right hand side h is an ordinary, albeit discontinuous, function, we can write it as an ordinary ...

    Solution Summary

    The expert explains a bit of distribution theory while explaining the problem. Two functions are examined.