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I have been trying to learn dirac delta function. but its kind of confusing. I come across 2 contrasting definitions for it. The first one states that the function delta(x-xo) is infinite at x=x0 while the other states that delta(x-x0) tends to infinite as x tends to x0. Now both of them are different and i'm not sure which one of the two is correct

Also alternatively, the integral of the delta function from -infinity to +infinity is 1. Now in that case since the value is 0 everywhere except x0, does this means the integral of the delta function between an interval (a b) which contains the x0 is also 1?

Also the second definition of delta function is integral between -infinity to +infinity F(x')delta(x-x')dx' is F(x). Can i derive this second definition from the above definitions? Because if its consistent, i must be able to derive this from the previous definitions right?

I would be happy if somebody could throw light on this question of mine