Found it:Let I be a real interval. Let a , b ∈ I such that ( a . . b ) is an open interval. Let f : I → R be a real function which is continuous on ( a . . b ). Let k ∈ R lie between f ( a ) and f ( b ). That is, either f ( a ) &lt; k &lt; f ( b ) or f ( b ) &lt; k &lt; f ( a ). Then ∃ c ∈ ( a . . b ) such that f ( c ) = k.Shouldn't this suppose continuity on the closed interval [ a , b ]? Otherwise consider f : [ − 1 , 1 ] → R such that f ( x ) = { arcsin ⁡ ( x ) -1&lt;x&lt;1 23 if x=1 − 23 if x= -1 where a = − 1 and b = 1.

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Found it:Let   I be a real interval. Let   a  ,  b  ∈  I such that   (  a  .  .  b  ) is an open interval. Let   f  :  I  →      R   be a real function which is continuous on   (  a  .  .  b  ). Let   k  ∈      R   lie between   f  (  a  ) and   f  (  b  ). That is, either   f  (  a  )  &amp;lt;  k  &amp;lt;  f  (  b  ) or   f  (  b  )  &amp;lt;  k  &amp;lt;  f  (  a  ). Then   ∃  c  ∈  (  a  .  .  b  ) such that   f  (  c  )  =  k.Shouldn&#039;t this suppose continuity on the closed interval   [  a  ,  b  ]? Otherwise consider   f  :  [  −  1  ,  1  ]  →      R   such that  f  (  x  )  =      {                            arcsin          ⁡          (          x          )                          -1&amp;lt;x&amp;lt;1                                      23                          if x=1                                      −          23                          if x= -1                        where   a  =  −  1 and   b  =  1.

plomet6a · Accepted Answer

Yes, I agree. The given statement is incorrect and your counterexample shows it.An even simpler example might just be  f  (  x  )  =      {                            0          ,                          x          =          −          1                                      2          ,                          −          1          &amp;lt;          x          &amp;lt;          1                                      1          ,                          x          =          1                        with   k  =      1    2  .

Let I be a real interval. Let a,b in I such that (a..b) is an open interval. Let f:I->R be a real function which is continuous on (a..b). Let k in R lie between f(a) and f(b). That is, either f(a)<k<f(b) or f(b)<k<f(a). Then EEc in (a..b) such that f(c)=k.

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New Questions in Calculus 2