A paper on calculus

Based on the differential mean value theorem and integral mean value theorem, this paper discusses the theoretical system of calculus, and especially proves the equivalence of three properties of continuous function and real number continuity in closed interval.

Keywords: real number continuity theorem; Equal value

Under the condition that f ′ (x) = f (x) is continuous in the closed interval [a, b], the contradiction between the differential of F(x) and the integral of f (x) can be revealed as a unity through the differential mean value theorem and the integral mean value theorem, thus establishing the basic theorem and formula of real one-variable function calculus. So how are these two mean value theorems established? We seek the source along the wave and get the theoretical system of real analysis, which is some theorems describing the continuity of real numbers, that is, the theoretical source of real analysis. The differential mean value theorem can be deduced from the lower theorem (see reference (1)).

Theorem 1 If f (x) is continuous on [a, b], then f (x) must have upper and lower bounds on [a, b]. This theorem can be derived from the following theorem.

Theorem 2 If f (x) is continuous in [a, b], then f (x) is uniformly continuous in [a, b].

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