[tex]r(x) = x[/tex] if [tex]x \geq 0[/tex] and [tex]r(x) = 0 [/tex] if [tex] x<0[/tex](adsbygoogle = window.adsbygoogle || []).push({});

I have to show that:

1-[tex]\[ \int_{- \infty}^{+ \infty} r(x) \varphi ''(x) dx = \varphi(0) \][/tex]

And 2- that the second derivative of r is the Dirac delta.

And I managed to do this by integrating by parts. Howver, I don't get why I can't just write:

[tex]\[ \int_{- \infty}^{+ \infty} r''(x) \varphi (x) dx = \varphi(0) \][/tex]

Wouldn't that be correct considering distributions (I actually used this to show the second point)? I guess my question is, why can't I write the second derivatives of the ramp function (the derivative of the Heaviside function) simply as

[tex]r(x) = 0[/tex] if [tex]x \geq 0[/tex] and [tex]r(x) = 0 [/tex] if [tex] x<0[/tex]

i.e. 0

Which would make the integral = 0

Does it only have a second derivative in the distribution sense? Why?

EDIT: I don't get why my message is being displayed like this...

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# Homework Help: Ramp function, Dirac delta function and distributions

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