Intermediate value theorem

The intermediate value theorem says that if a function, formula_1, is continuous over a closed interval formula_2, and is equal to formula_3 and formula_4 at either end of the interval, for any number, c, between formula_3 and formula_4, we can find an formula_7 so that formula_8. 
This means that if a continuous function's sign changes in an interval, we can find a root of the function in that interval. For example, if formula_9 and formula_10, we can find an formula_7 in the interval formula_12 that is a root of this function, meaning that for this value of x, formula_13, if formula_1 is continuous. This corollary is called Bolzano's theorem.