Suppose I have two continuous random variables $X, Y$ with joint probability density $f(x,y)$. Then by marginalizing over $y$ I can derive
$ f(x| Y=y) \\ = \int{f(x, \hat{y}|Y=y) d\hat{y}} \\ = \int_{\hat{y} \neq y}{f(x, \hat{y}|Y=y) d\hat{y}} \\ = \int_{\hat{y} \neq y}{0 \, d\hat{y}} \\ = 0 $
where I'm using the fact that it is possible to exclude single points from an integral without changing its value.
Now this is obviously not correct, but since my probability theory courses were more pragmatic than rigorous, I'm not sure where exactly the error is.
*Edit: * After Didier's question, I think I can answer this myself: The joint distribution $f(x,y,\hat{y})$ should intuitively be $f(x,y)\delta(y-\hat{y})$. Since this is not a conventional function, the usual rule "it is possible to ignore single points of an interval" does not apply any more. Correct?