Let be any 1-variable function. Define and .
Prove that is continuous and are closed in .
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Use the sequence definitions! Keep in mind a sequence of coordinate pairs converges if and only if the individual coordinates converge, and relate that to the convergence-preserving properties of continuous functions.
where do you want a sequence to converge?
Originally Posted by Kat-M where do you want a sequence to converge? Formally, here are the two qualities I was talking about:
Let be a sequence of points in . Then if and only if and .
Let . Then is continuous at if and only if for every sequence , .
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