The statement for Intermediate Value Theorem is thus:Let f denote a function that is continuous on the closed interval [ a , b ] and suppose f ( a ) ≠ f ( b ). If N is any number between f ( a ) and f ( b ), then there is at least one number c in ( a , b ) so that f ( c ) = N.But who is to stop me if I put it like this?Let f denote a function that is continuous on the open interval ( a , b ). Now, lim x → a + f ( a ) = L and lim x → b − f ( b ) = M. Suppose, L ≠ M. If N is any number between L and M, then there is at least one number c in ( a , b ) so that f ( c ) = N.

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The statement for Intermediate Value Theorem is thus:Let   f denote a function that is continuous on the closed interval   [  a  ,  b  ] and suppose   f  (  a  )  ≠  f  (  b  ). If   N is any number between   f  (  a  ) and   f  (  b  ), then there is at least one number   c in   (  a  ,  b  ) so that   f  (  c  )  =  N.But who is to stop me if I put it like this?Let   f denote a function that is continuous on the open interval   (  a  ,  b  ). Now,       lim          x      →              a        +              f  (  a  )  =  L and       lim          x      →              b        −              f  (  b  )  =  M. Suppose,   L  ≠  M. If N is any number between   L and   M, then there is at least one number   c in   (  a  ,  b  ) so that   f  (  c  )  =  N.

ontzeidena8a · Accepted Answer

Nobody.In fact, with   f as in your suggestion, define   g  :  [  a  ,  b  ]  →      R  ,  g  (  x  )  =      {                            f          (          x          )                          a          &amp;lt;          x          &amp;lt;          b                                      L                          x          =          a                                      M                          x          =          b                        Then the classical IVT applied to   g is equivalent to your variant applied to   f. With this in mind, the standard formulation looks perhaps more appealing.

Jorge Schmitt · Accepted Answer

Your version follows from the previous version. Let   F  (  x  )  =  f  (  x  ) for   a  &amp;lt;  x  &amp;lt;  b,   F  (  a  )  =  L and   F  (  b  )  =  M. Then   F is continuous on   [  a  ,  b  ] and the first version applied to   F gives you the new version.

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