Intermediate value theorem: also known as intermediate value theorem, it is one of the properties of continuous function on closed interval and one of the important properties of continuous function on closed interval. In mathematical analysis, the intermediate value theorem shows that if the domain is a continuous function f of [a, b], that is to say, the function value in an interval of the intermediate value theorem is a continuous function must be between the maximum value and the minimum value.

Zero theorem: If the image of the function y= f(x) in the interval [a, b] is a continuous curve and has f (a) f (b).

Extended data

Proof of Zero Theorem: Let F (a)

From f (a) < 0, we know that e ≠ φ and b are an upper bound of e, so according to the existence principle of supremum, there exists ξ=supE∈[a, b].

F(ξ)=0 (note that f(a)≠0 and f(b)≠0, so there must be ξ∈(a, b) at this time).

In fact,

(i) If f (ξ) is known from the local sign-preserving property of function continuity; 0, for x 1 ∈ (zeta, zeta+δ): f (x) SupE, the upper bound contradiction with SupE is e;

(ii) if f (ξ) >; 0, then ξ∈(a, b]. The existence of δ > is still known from the local sign-preserving property of function continuity. 0, there is an upper bound for x 1∈(ξ-δ, ξ): f (x) > 0→, where x 1 is e and x 1

Combining (i) and (ii) means that f(ξ)=0.

We can also use the closed interval set theorem to prove the zero point theorem.

Baidu Encyclopedia-Intermediate Value Theorem

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