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.
Show that the set $\{x\in[0,1]\mid f(x)=g(x)\}$ is compact when $f,g$ are continuous
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real-analysis
continuity
compactness
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0The 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. – 2012-12-17
1 Answers
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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.