Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval [a, b] and s {\displaystyle s} is a number such that f ( a ) < s < f ( b ) {\displaystyle f(a)<s<f(b)} , then there exists some x {\displaystyle x} between a {\displaystyle a} and b {\displaystyle b} such that f ( x ) = s {\displaystyle f(x)=s} . That is, the image of a continuous function over an interval is itself an interval that contains f ( a ) , f ( b ) {\displaystyle f(a),f(b)} . For example, suppose that f ∈ C ( [ 1 , 2 ] ) , f ( 1 ) = 3 , f ( 2 ) = 5 {\displaystyle f\in C([1,2]),f(1)=3,f(2)=5} , then the graph of y = f ( x ) {\displaystyle y=f(x)} must pass through the horizontal line y = 4 {\displaystyle y=4} while x {\displaystyle x} moves from 1 {\displaystyle 1} to 2 {\displaystyle 2} . Over the interval, the set of function values has no gap, and the graph can be drawn without lifting a pencil from the paper. The corollary Bolzano's theorem states that if a continuous function has values of opposite sign inside an interval, then it has a root in that interval. The theorem depends on, and is equivalent to, the completeness of the real numbers, although Weierstrass Nullstellensatz is a version of the intermediate value theorem for polynomials over a real closed field. A similar result to the intermediate value theorem is the Borsuk–Ulam theorem, which underpins why rotating a wobbly table will always bring it to stability. Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value...