consider a function . Prove that f is continuous on R iff for every subset B of Real
I think you meant:
I is continuous, then for any open set is an open set. So let be an element of i.e. there exists an open set such that Then is an open subset of and contains therefore and we can conclude: continuous
Conversely, assume f . Let be an open set. Then intU=U, and we can write:
But, for any subset means that is open (if you never saw that, try to prove it)
Thus for any open subset , is open so f is continuous, we just proved: continuous.