continuous on metric space

Let . Prove that f is continuous if and only if for every if and only if for every .
Give an example of a continuous f such that for some .

I have shown that the closure case is true, i need some help on the interior case and the example.
I was told to use the fact that inverse image of every open set is open

Quote:

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Originally Posted by wopashui

Let . Prove that f is continuous if and only if for every .

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Note that if is an open set in then .
If then