I am not exactly sure what you mean by Cauchy law.
I think what you are using here is the intermediate value theorem.
Since, and
It means by the intermediate value theorem there is so that
Thus, you need the condition is continous on .
i got this question
http://img412.imageshack.us/img412/3713/88436110xw9.gif
here is the solution:
they are taking the minimal value
and the maximal value
the innequalitty that the write is correct min< <max
but why??