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Mar 10, 2012 at 5:32 comment added Rami I did not explained why it is enough to show that $(x|g(x) < c)$ is Borel. I now understand that is probably the point that you where interested in. But it is explained in the answer of GH so there is no point to repeat it
Mar 10, 2012 at 5:23 comment added Rami In-fact the division of $U$ into 2 sets is unnecessary and the image of $U$ is just open. This dose not prove continuity yet since it is not enough to check continuity on sets like this.
Mar 10, 2012 at 5:03 history edited Rami CC BY-SA 3.0
added 6 characters in body
Mar 10, 2012 at 4:56 history answered Rami CC BY-SA 3.0