What is bolzanos theorem?

Bolzano's theorem, also known as the intermediate value theorem, is a concept in calculus that states that if a continuous function f(x) takes on two different values, let's say A and B, at two different points within a closed interval [a, b], then it must take on every value between A and B at least once within the interval [a, b]. This theorem is named after the Czech mathematician Bernard Bolzano, who introduced it in the early 19th century.

In simpler terms, Bolzano's theorem says that if a function is continuous and its graph starts at one value and ends at another, it must pass through every value in between. This theorem is a fundamental tool in calculus and is used to prove many other theorems and concepts, as well as in applications such as optimization and engineering.