This theorem fails to be true for the rationals because they are not con- nected; for example, although there exist rationals a,b such that a2 = 0 and b2 = 4, there exists no rational c such that c2 = 2.
Simon Stevin proved this result for polynomials by constructing the decimal expansion of a solution, and a function is said to be Darboux if it ful lls the property laid out in this result. Bolzano's theorem is equivalent to this result, which is true because continuous functions take connected sets to connected sets. This theorem fails to be true for the rationals because they are not con- nected; for example, although there exist rationals a,b such that a2 = 0 and b2 = 4, there exists no rational c such that c2 = 2. For ten points, identify this result from real analysis stating that any function from the interval [a,b] to the real line must assume every value between f(a) and f(b). 
Simon Stevin proved this result for polynomials by constructing the decimal expansion of a solution, and a function is said to be Darboux if it ful lls the property laid out in this result.
intermediate value theorem [accept Bolzano's theorem until  read] 