Prove the property of Dirac's Delta

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Let \lbrace x_i \rbracebe the zeroes of function f(x). Near the zeroes, you can write the function as f(x) = f'(x_i)(x-x_i) + O(x-x_i)^2. As the Dirac delta is entirely localized at xi, you can just drop all higher order terms and it's still exact. Then you can use the scaling property of delta function, http://en.wikipedia.org/wiki/Dirac_delta_function#Scaling_and_symmetry
 
Ok. Thank you clamtrox.
 
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