If f : R → R is continuous over the entire domain Ms is not bounded above or below, show that f ( R ) = R .My approach would be to say that since f is continuous on R it must be continuous on [ a , b ] , a , b ∈ R and using the intermediate value theorem this tells us that f must take every value between f ( a ) and f ( b ). Now I want to argue that we can make this closed bounded interval as large as we want and the result still holds and I want to show that this implies that we can make the values of f ( a ) as small as we like and f ( b ) as large as we like (or the other way round) since f is neither bounded below or above but I'm not sure if this argument can used to show that the image of f is R since we switch from closed bounded intervals to an unbounded interval.Does my argument hold to show the result?

Question

If   f  :      R    →      R   is continuous over the entire domain Ms is not bounded above or below, show that   f  (      R    )  =      R    .My approach would be to say that since   f is continuous on       R   it must be continuous on   [  a  ,  b  ]  ,     a  ,  b  ∈      R   and using the intermediate value theorem this tells us that   f must take every value between   f  (  a  ) and   f  (  b  ). Now I want to argue that we can make this closed bounded interval as large as we want and the result still holds and I want to show that this implies that we can make the values of   f  (  a  ) as small as we like and   f  (  b  ) as large as we like (or the other way round) since   f is neither bounded below or above but I&#039;m not sure if this argument can used to show that the image of   f is       R   since we switch from closed bounded intervals to an unbounded interval.Does my argument hold to show the result?

Sanaa Hudson · Accepted Answer

A little bit cleaner argument:Suppose   c  ∈      R  . We will show that there exists   x  ∈      R   with   f  (  x  )  =  c  f is not bounded below, so there exists   a  ∈      R   such that   f  (  a  )  &amp;lt;  c  f is not bounded above, so there exists   b  ∈      R   such that   f  (  b  )  &amp;gt;  cNow   f is continuous on   [  a  ,  b  ] (or   [  b  ,  a  ] if   b  &amp;lt;  a) and   c  ∈  (  f  (  a  )  ,  f  (  b  )  ), so by intermediate value property there exists   x  ∈  (  a  ,  b  ) (or   (  b  ,  a  )) with   f  (  x  )  =  c

If f:RR->RR is continuous over the entire domain Ms is not bounded above or below, show that f(RR)=RR.

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