Suppose you have the following definition of Dirac-delta function, or the so called sifting property:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \int^{d}_{c}f(t)δ(t-a)dt =\left\{\begin{array}{cc}f(a),&\mbox{ if }

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. [/tex]

Can I use this to prove the following?

[tex] \mbox{ 1) } δ(t) = 0 \ t ≠ 0 [/tex]

[tex] \mbox{ 2) } δ(t) \mbox{ is undefined at } t = 0 [/tex]

[tex] \mbox{ 3) } \int^{∞}_{-∞}δ(t)dt = 1 [/tex]

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

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# Using sifting property to prove other properties

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