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

clovnerie0q 2022-09-30 Answered
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?
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Answers (1)

Sanaa Hudson
Answered 2022-10-01 Author has 7 answers
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 ) < c
f is not bounded above, so there exists b R such that f ( b ) > c
Now f is continuous on [ a , b ] (or [ b , a ] if b < a) and c ( f ( a ) , f ( b ) ), so by intermediate value property there exists x ( a , b ) (or ( b , a )) with f ( x ) = c
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