Show that the function f defined by f(x)=0 if x is irrational and f(x)=1 if x is rational does not have a limit at any point. Any suggestions?
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Originally Posted by paperclip Show that the function f defined by f(x)=0 if x is irrational and f(x)=1 if x is rational does not have a limit at any point. Any suggestions? Let for all , then we know that for all and as . Therefore, we have for all but . Thus is not continuous at . However, you can give a good proof for the general case since for any , you may find and such that as . I hope this can help you...
Originally Posted by paperclip Show that the function f defined by f(x)=0 if x is irrational and f(x)=1 if x is rational does not have a limit at any point. Any suggestions? You can solve this using the facts that: (a) for any (b) for any , there exists a sequence such that and That is to say, for any irrational number there is a sequence of rationals that converges to it.
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