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The resulting limit is the [[conditional probability distribution]] of {{mvar|Y}} given {{mvar|X}} and exists when the denominator, the probability density <math>f_X(x_0)</math>, is strictly positive.
It is tempting to ''define'' the undefined probability <math>P(A \mid X=x)</math> using limit ({{EquationNote|1}}), but this cannot be done in a consistent manner. In particular, it is possible to find random variables {{mvar|X}} and {{mvar|W}} and values {{mvar|x}}, {{mvar|w}} such that the events <math>\{X = x\}</math> and <math>\{W = w\}</math> are identical but the resulting limits are not:
:<math>\lim_{\epsilon \to 0} P(A \mid x-\epsilon \le X \le x+\epsilon) \neq \lim_{\epsilon \to 0} P(A \mid w-\epsilon \le W \le w+\epsilon).</math>
The [[Borel–Kolmogorov paradox]] demonstrates this with a geometrical argument.
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