Continuous functions

dwsmith

MHF Hall of Honor
Mar 2010
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Florida
Which of the following statements are true about the open interval (0,1) and the closed interval [0,1]?

1. There is a continuous function from (0,1) onto [0,1].
2. There is a continuous function from [0,1] onto (0,1).
3. There is a continuous one-to-one function from (0,1) onto [0,1].

How to prove or disprove each choice?
 
Nov 2009
485
184
Maybe I can give a a hint: continuous functions preserve compactness.