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Sets and graphs

There are a couple of concepts I need clarification for:

1) If a set has no interior points, then is it necessarily closed? Isn't the empty set considered open?

2) If the graph of f: R -> R is connected, does it have to be continuous? In just Real -> Real aren't these definitions equivalent?


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1) If a set has no interior points, it is not necessarily closed.
For example, X={1/n} is a set with no interior points, ...

Solution Summary

This helps clarify open and closed sets, and connected and continuous graphs.