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Let C be the Cantor set defined C=intersection sign on top inf bottom n=0 C_n.Define g:[0,1]->R by g(x)={1 if x belong to C and 0 if x does not belong to C.
a-show that g fails to be continuous at any point c belong to C.
b-prove that g is continuous at every point c does not belong to C.

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Solution Summary

This is a proof regarding continuity and belonging to sets.

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Proof:

We know that the Cantor set C is a closed set with no interior points. So [0,1]-C is an open set.

a. For each point x in C, because x is not an interior point, then in any neiborhood ...

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