Ramp function, Dirac delta function and distributions

[Amok]

r(x) = x if x \geq 0 and r(x) = 0 if x<0

I have to show that:

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

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:

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

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

r(x) = 0 if x \geq 0 and r(x) = 0 if x<0

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...

 

[vela]

Change the backslash in your closing tex tags to a forward slash, i.e. /tex instead of \tex.