Dirac Delta Function: Definition & Mathematics

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Vahsek
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It's been quite some time now since I decided to stop self-studying physics and to pay more attention to the math behind. I'm working towards gaining an understanding of 100% rigorous mathematics for now.

One thing that has always bothered me is the Dirac delta function. What I want to know is the following: is the dirac delta function rigorously defined in any area of mathematics or is it just some kind of "fuzzy" mathematics to help with computations? Secondly, if it is rigorously defined somewhere, in which topic of mathematics is it? (real analysis, complex analysis, functional analysis... ?)
 

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