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griffiths said:

How is it "nothing but the Dirac delta function"?? (which is not even a function).

Couldn't [itex]g_{y}(x)[/itex] simply be a function like (for any constant y)

[itex]g_{y}(x)[/itex] = 1 | x=y

[itex]g_{y}(x)[/itex] = 0 | elsewhere

Then we have, [itex]x * g_{y}(x) = y*g_{y}(x)[/itex], so it is indeed an eigenfunction of the x operator. And it happens to be a normal function. So why does he say it's 'nothing but Delta'?