Let be continuous with .
If for every the limits and have ,
then there exists some such that contains both a positive and negative (but possibly infinite) number. #m/thm/anal
Proof
By the extreme value theorem reaches either its maximum or minimum at ,
for if both lay on the boundary then would be constant.
Without loss of generality assume has a maximum at (otherwise consider ).
Clearly
and similarly
as required.
In case is differentiable, Rolle's theorem is equivalent to saying there exists some such that ,
which itself is a special case of the mean value theorem.