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I am not a mathematician so my question might be silly.

I really came across it in physics but I think it is purely mathematical:

I came across an equation of the form:

delta(m-n)*A= delta(m-n)*B

my question is now for what cases can I conclude A=B?

Does this only hold for m=n, or can I also conclude A=B for m->n ,i.e. m=n+epsilon for sufficiently small epsilon. I am not sure about this as the naive definition of the delta function(infinite at 0; 0 everywhere else and integrates to 1 when 0 is in the integration range) would lead me to conclude that only when m=n I could conclude A=B. However the delta function is the limit of a set of function which do not vanish for for m=n+epsilon so maybe their is an argument why for epsilon small enough I should be able to conclude A=B. Another reason why I think this should be the case is that this argument came up in a paper of a nobel laureate and his argument crucially depended on the possibility to conclude A=B even when m not equal n but only m=n+epsilon. So if anybody could give me a rigorous argument why this should be possible I would be thankful to hear about it.