Piecewise smooth functions..

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- Oct 15th 2012, 01:12 PMsarideli18P.S
Piecewise smooth functions..

- Oct 15th 2012, 02:25 PMTheEmptySetRe: I need help with piecewise smooth functions
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.

This limit can be resolved using L'hosipitals

by L.H

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

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. - Oct 15th 2012, 02:46 PMsarideli18Re: 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.

- Oct 15th 2012, 02:49 PMsarideli18Re: 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.

- Oct 15th 2012, 02:57 PMTheEmptySetRe: I need help with piecewise smooth functions
- Apr 13th 2013, 12:41 AMaddictionsharyRe: 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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