Prove that the function f defined by f(x)=x if x is rational and f(x)=-x if x is irrational is continuous at 0 only.
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Prove that the function f defined by f(x)=x if x is rational and f(x)=-x if x is irrational is continuous at 0 only.
Because this is not a homework service, you need to show us some effort on your part.Quote:
Originally Posted by redsoxnation
We will happily give you help on what you have already done.
But do not expect us to do your work for you.
Hint: If f is indeed continuous, then f^{-1}( (0,1) ) is a non-empty open subset of R. It also contains 1/2, for instance. Since 1/2 lies in an open subset of R, look at an irrational number in a neighborhood of 1/2 and show that it doesn't map into (0,1) as originally assumed.Quote:
Originally Posted by redsoxnation
I DO NOT want to do your HW for you...Plato is right, show us some work. But, a hint that may be easier than patrick is to consider thatQuote:
Originally Posted by redsoxnation
is continuous implies that