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Illustration Dirac's delta function picks the function value at the origin

Dirac's delta function picks the function value at the origin Delta function picks the function value at the origin in an interval
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Dirac's delta function \( \delta(x)\) picks in an integral the function value \(f(0)\) at the origin \( x = 0 \):$$ \int_{-\infty}^{\infty} f(x) \, \delta(x) ~ \text{d}x ~=~ f(0) $$

If the integral is considered in the interval between \(a\) and \(b\):$$ \int_{a}^{b} f(x) \, \delta(x) ~ \text{d}x ~=~ f(0) $$then \(x=0\) must lie in this interval, otherwise the integral is zero.