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Impuse Function Property

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magnetra

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I have an elementary question in Signal Analysis. The unit impulse function has this property
x(t).δ(t) = x(0).δ(t) ---------(1)

∫x(t).δ(t)dt = x(0) ----------(2)

How are these two expression different? I think eq (1) should be equal to x(0).
Please clarify me on this.
Regards
M
 

You have probably answered your own question by now but either way I think you are right.
 

In equation (1) x(0)δ(t) is not equal to x(0). because x(0)δ(t) has only a single point at zero with value x(0). But x(0) has infinite values all equal to x(0) for every point. and

∫x(t)δ(t)dt=∫x(0)δ(t)dt=x(0)∫δ(t)dt=x(0)
 
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