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Mathematical analysis (1) proof of finite covering theorem
The key is to prove that f(I) is intermediate. In fact, this problem is to prove the intermediate theorem of continuous functions.

If it is an open interval, it can be said that the function is extended to a closed interval, and the endpoint function value can take the corresponding unilateral limit.

In addition, if it is an unbounded interval, let it be [a, +∞), as long as it is proved that for any m >;; A, both f([a, M]) are intervals.

In fact, this problem boils down to proving the intermediate value theorem of continuous functions on a closed interval by using finite covering theorem, and only need to prove the zero point theorem, that is, if f∈C[a, b] and f (a) f (b).

If the conclusion is not true, then for any x_{0}, there exists f(x0)≠0. According to the local sign-preserving property of continuous function, there is a neighborhood U0 corresponding to x0, which makes f sign-preserving on U0. Therefore, when x0 traverses all the points in [a, b], we get the open cover of [a, b].

U0,U 1,…

(Unilateral interval at the endpoint) Therefore, there are a finite number of V 1, V2, ..., vk covers [a, b], and f is sign-preserving on every Vi.

Moreover, each Vi must intersect with another (the concept of Lebesgue number), so it is obvious that the signs of F (a) and F (b) are the same, which contradicts the hypothesis.