Intermediate value theorem

Continuous function on an interval takes on every value between its values at the ends

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} .

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