I found that f ( x ) = x 2 is continuous on [ 1 , 2 ] but does not assume the values 2 intermediate between the value 1 and 4. I do not understand what they mean. For me, 2 ∈ [ 1 , 2 ] and f ( 2 ) = 2.Also, using the the intermediate value theorem, since f is continuous on [ 1 , 2 ] and f ( 1 ) = 1 &lt; 2 &lt; 4 = f ( 2 ), there exists c ∈ [ 1 , 2 ] such that f ( c ) = 2.Can I get some explanation about this?

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I found that   f  (  x  )  =      x    2   is continuous on   [  1  ,  2  ] but does not assume the values 2 intermediate between the value 1 and 4. I do not understand what they mean. For me,       2    ∈  [  1  ,  2  ] and   f  (      2    )  =  2.Also, using the the intermediate value theorem, since   f is continuous on   [  1  ,  2  ] and   f  (  1  )  =  1  &amp;lt;  2  &amp;lt;  4  =  f  (  2  ), there exists   c  ∈  [  1  ,  2  ] such that   f  (  c  )  =  2.Can I get some explanation about this?

embutiridsl · Accepted Answer

Looking at the text before that series of examples, the authors specifically say:We conclude this chapter with a collection of functions defined on a closed interval   [  a  ,  b  ]  ⊂      Q   and having values in       Q  .So, we&#039;re not in       R   anymore.

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