The Core Content and Proof Strategy of the Original Theorem

Introduction

The Original Theorem, a cornerstone of modern calculus, establishes a fundamental relationship between differentiation and integration. It asserts that if a function $ f(x) $ is continuous on the closed interval $[a, b]$, then the integral of $ f(x) $ over $[a, b]$ is differentiable, and its derivative equals $ f(x) $. This theorem, often referred to as the Fundamental Theorem of Calculus (FTC), bridges the gap between the two foundational concepts in analysis. Its proof strategy relies on the interplay of continuity, the Mean Value Theorem for integrals, and the Mean Value Theorem for derivatives. By systematically dissecting the theorem’s components and the logical steps of its proof, this article explores the core content and the strategic approach that underpins its validity.

Core Content

Statement of the Theorem

The Original Theorem states that for a function $ f(x) $ defined on $[a, b]$ and continuous on $[a, b]$, the function $ F(x) = \int_a^x f(t), dt $ is differentiable on $[a, b]$, and its derivative satisfies $ F'(x) = f(x) $. This result is pivotal in calculus, as it allows the computation of integrals through the evaluation of antiderivatives. The theorem’s validity hinges on the continuity of $ f(x) $, which ensures the existence of the integral and the differentiability of $ F(x) $.

Key Components of the Theorem

The theorem’s core lies in three interdependent components:

  1. Continuity of $ f(x) $: The function $ f(x) $ must be continuous on $[a, b]$ to guarantee the existence of the integral $ F(x) $.
  2. Differentiability of $ F(x) $: The integral $ F(x) $ is differentiable on $[a, b]$, and its derivative coincides with $ f(x) $.
  3. Integral as a Function: The integral $ F(x) $ is a function that encapsulates the area under $ f(x) $, and its derivative reflects the instantaneous rate of change of this area.

These components are not isolated but are connected through the theorem’s proof strategy, which ensures that the continuity of $ f(x) $ guarantees the differentiability of $ F(x) $.

Proof Strategy

The proof of the Original Theorem employs a sequence of logical steps rooted in the properties of continuous functions and the Mean Value Theorem for integrals.

Step 1: Establishing Differentiability

To demonstrate that $ F(x) $ is differentiable, the proof begins by leveraging the Mean Value Theorem for integrals. This theorem states that if $ f(x) $ is continuous on $[a, b]$, there exists a point $ c \in [a, b] $ such that $ \int_a^b f(t), dt = f(c)(b - a) $. This result ensures that the integral $ F(x) $ is well-defined and that its value is directly tied to the function $ f(x) $.

Step 2: Applying the Mean Value Theorem for Derivatives

The next critical step involves the Mean Value Theorem for derivatives. This theorem asserts that if $ f(x) $ is continuous on $[a, b]$ and differentiable on $ (a, b) $, there exists a point $ c \in (a, b) $ such that $ F'(c) = f(c) $. The theorem’s application here relies on the continuity of $ f(x) $, which ensures that the derivative of $ F(x) $ exists and equals $ f(x) $ at every point in $[a, b]$.

Step 3: Formalizing the Derivative

The proof concludes by formally establishing the relationship between $ F(x) $ and $ f(x) $. By combining the results from the Mean Value Theorem for integrals and derivatives, the theorem demonstrates that the derivative of the integral of $ f(x) $ is precisely $ f(x) $. This connection is crucial for the theorem’s validity, as it confirms that