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We know that $$ \int_{\mathbb{R}} f(t)\delta(t) \mathrm{d}t = f(0) $$ if $f$ is continuous. What will it be if $f$ is not continuous? For instance, what is the value of $$ \int_{\mathbb{R}} e^t\mathrm{u}(t)\delta(t) \mathrm{d}t $$

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Typical strategy: Apply Laplace transform. –  Eric Towers Mar 20 at 23:50
Can you please elaborate? –  Priyatham Mar 20 at 23:53
Dirac delta is linear operator acting on the space of continuous functions, not the integral against something. You might try to approach $\delta$ by a sequence of $C^\infty_c$ functions in the sense of distributions and then try look what it gives when applied to your test function. –  TZakrevskiy Mar 21 at 9:26
That doesn't help as the answer depends on the particular function I choose to approach the $\delta$ function. –  Priyatham Mar 21 at 16:17

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