Prove or Disprove: Suppose that f is a real-valued function that is continuous on a nonempty set S in R n and that f ( S ) is compact in R . Then S is a compact set R n .I just studied the concept of compactness and I am stuck trying to prove/disprove the above statement. Any help is appreciated.

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Prove or Disprove: Suppose that   f is a real-valued function that is continuous on a nonempty set   S in             R        n   and that   f  (  S  ) is compact in       R  . Then   S is a compact set             R        n  .I just studied the concept of compactness and I am stuck trying to prove/disprove the above statement. Any help is appreciated.

Ari Jacobs · Accepted Answer

Let   f  :      R    →      R   be a continuous function which is 1 if   x is in   (  −  1  ,  1  ), 0 outside of   [  −  2  ,  2  ], and between 0 and 1 on the intervals   [  −  2  ,  −  1  ] and   [  1  ,  2  ]. Then   f  (  (  −  1      /    2  ,  1      /    2  )  )  =  {  1  } but   (  −  1      /    2  ,  1      /    2  ) is not compact.

Prove or Disprove: Suppose that f is a real-valued function that is continuous on a nonempty s

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