What is Bolzano's theorem?
In the usage I am familiar with: If a function is continuous on an interval and is sometimes positive and sometimes negative on the interval, then the function has a zero in the interval.
Bolzano's Theorem is a special case of the Intermediate Values Theorem.
There is also a Bolzano-Weierstrass Theorem (Which states that every bounded infinite set of real numbers has an accumulation point.)
Different communities pf mathematicians sometimes use different names for the same theorem or the same names for different theorems. In my limited experience, i haven't encountered a different "Bolzano Theorem".