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I am trying to show that the set $S=\{x\in[0,1]\mid f(x)=g(x)\}$ is compact given that function f(x) and g(x) is continuous and the set $U=\{x\in(0,1)\mid f(x)>g(x)\}$ is open. I have no clue to approach.

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I have to admit that the reason to edit and remove "compact" and "open" is beyond me. – Asaf Karagila Dec 16 '12 at 23:09
Okay, I really have to ask. What was wrong with the edit where I just copied the image into text? Now the question looks jumbled up, and makes little sense. – Asaf Karagila Dec 16 '12 at 23:53
Oh, gosh, I read the new user's question posting guide and it said one shouldn't use "show that", "prove that" as it being impolite. – user48601 Dec 17 '12 at 0:10
The question was originally okay, I just replaced the image file with text. Then you began editing it... and now the question is less clear than before. I would suggest rolling back to my edit. Not to mention the tag changes, which were completely unnecessary. – Asaf Karagila Dec 17 '12 at 0:12
up vote 3 down vote accepted

Hint: Recall that $f-g$ is a continuous function. For the first show that $S$ is the preimage of a closed set, intersected with a compact set; for the second show that $U$ is the preimage of an open set, intersected with an open interval.

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