Let f(x) be a real-valued function that is continuous on the closed interval [a, b]. Then for every value y_0 between f(a) and f(b), there is a value x_0 in [a, b] such that f(x_0) = y_0, meaning that f(x) assumes all intermediate values between f(a) and f(b).

closed interval | continuous function

Rolle's theorem

mean value theorem | Bolzano theorem

Jean-Gaston Darboux | Augustin-Louis Cauchy | Joseph-Louis Lagrange | Bryson of Heraclea