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Math Help - I need help with piecewise smooth functions

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    Piecewise smooth functions..
    Last edited by sarideli18; October 15th 2012 at 07:38 PM.
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    Re: I need help with piecewise smooth functions

    Quote Originally Posted by sarideli18 View Post
    Indicate which of the following functions are piecewise smooth on [-1,1] and why:

    i.) h(x)=x^2ln|x|

    ii.) q(x)=xln|x|

    iii.) r(x)= 2/(1-x) if x≠1 and 0 if x=0

    iv.) s(x)= 1/(2-x) if x≠1 and 0 if x=0
    The definition of peicewise smooth is that a function is differentable on the domain except at a finite number of points. Also at these points both the left and right derviatve must exist.

    So for i)

    h(x) is continous and differentable everywhere except at x=0. For simplicity note that h(x) is an even function so we only have to test the derviative on one side.

    \lim_{h \to 0}\frac{(0+h)^2\ln(0+h)}{h} = \lim_{h \to 0}h\ln(h)

    This limit can be resolved using L'hosipitals
    by L.H

    \frac{\ln(h)}{\frac{1}{h}}

    So now this is the form infinity divided by infinity so we take the derivative to get

    \frac{\frac{1}{h}}{\frac{-1}{h^2}}=-h

    So the limit is 0 and both the left and right derviative exist at that point. Also the function can be made continous if we define h(0)=0.

    Now try the 2nd one.
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    Re: I need help with piecewise smooth functions

    Well, I just found that the first and fourth ones are piecewise smooth, and the second one is not piecewise smooth. But I am stuck on the third one. Since the function itself (2/(1-x)) and 0) and its derivative (2/(x-1)^2 and 0) are continuous I think it is piecewise smooth. But there is a jump at x=1. That confuses me.
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    Re: I need help with piecewise smooth functions

    I mean, the limit from left is infinite and the limit from right is 0. It is the same case as the derivative of the function. Therefore, I think it is not piecewise smooth, but I am not sure.
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    Re: I need help with piecewise smooth functions

    Quote Originally Posted by sarideli18 View Post
    I mean, the limit from left is infinite and the limit from right is 0. It is the same case as the derivative of the function. Therefore, I think it is not piecewise smooth, but I am not sure.
    You are correct the derivative cannot be infinty at a point so it does not exist.
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    Re: I need help with piecewise smooth functions

    I just found that the first and fourth ones are piecewise smooth, and the second one is not piecewise smooth.

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