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*definition*of a conditional probability density? My course seems to state it as a theorem, without proof, but then again my course is a little bit vague; although I welcome replies on this part, this is not the essential of this topic)

Anyway, the confusion is the following: let the s.v. Y be the s.v. X, then of course [itex]f(X = a | X=b)[/itex] should be zero if a is not equal to b (if the expression means what it is meant to mean), however it is equal to [itex]\frac{f(X = a)}{f(X=b)}[/itex] and there's no real reason why this should be zero.