Using sifting property to prove other properties

[Bipolarity]

Suppose you have the following definition of Dirac-delta function, or the so called sifting property:

$$\int^{d}_{c}f(t)δ(t-a)dt =\left\{\begin{array}{cc}f(a),&\mbox{ if }<br /> c\leq x \leq d\\0, & \mbox{ if } x>d \mbox{ or } x<c \\ \mbox{undefined}, & \mbox {if } x = d \mbox{ or } x = c \end{array}\right.$$

Can I use this to prove the following?

$$\mbox{ 1) } δ(t) = 0 \ t ≠ 0$$
$$\mbox{ 2) } δ(t) \mbox{ is undefined at } t = 0$$
$$\mbox{ 3) } \int^{∞}_{-∞}δ(t)dt = 1$$

I was able to prove property 3 but it seems not possible to prove the first two. But I am probably mistaken else my text would not use the sifting property to define this unit impulse function. Any ideas? Thanks!

BiP

 

[Office_Shredder]

The delta function is rigorously defined as a linear functional on the vector space of all functions (which satisfy nice properties, e.g. continuous, decay quickly, etc.). If we started using that terminology would you be totally lost?

"Proving" that delta(0) is not a number is fairly simple in a naive way:
$$\int_{0}^{0} 1*\delta(t) dt = 1$$
If delta(0) was a number then we would get the answer 0, not 1, so delta(0) can't be a number

 

[Bipolarity]

I'm sorry Office I don't fully comprehend your proof. I don't understand why Dirac(0), if it were a number, would lead to a contradiction involving the integral in your post. My knowledge of Dirac delta is very sloppy, nor do I have any idea what a functional is. And any tips on how to prove property 1?

BiP