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Relation to completeness

The theorem depends on, and is equivalent to, the completeness of the real numbers. The intermediate value theorem does not apply to the rational numbers Q because gaps exist between rational numbers; irrational numbers fill those gaps. For example, the function f (x) = x 2 {\displaystyle f(x)=x^{2}} for x ∈ Q {\displaystyle x\in \mathbb {Q} } satisfies f (0) = 0 {\displaystyle f(0)=0} and f (2) = 4 {\displaystyle f(2)=4}. However, there is no rational number x {\displaystyle x} such that f (x) = 2 {\displaystyle f(x)=2}, because 2 {\displaystyle {\sqrt {2}}} is an irrational number.