# Thread: Fundamental Theorem of Calculus

1. ## Fundamental Theorem of Calculus

Theorem. Let $\displaystyle a \in \mathbb{R}$, and let $\displaystyle I$ be an interval containing $\displaystyle a$. Let $\displaystyle I \subseteq K \subseteq \mathbb{R}$. Suppose that the function $\displaystyle f: K \to \mathbb{R}$ is integrable on $\displaystyle [a,x]$ for all $\displaystyle x \in I$ with $\displaystyle x > a$ and on $\displaystyle [x,a]$ for all $\displaystyle x \in I$ with $\displaystyle x < a$. Let $\displaystyle F_a$ be the area accumulation function for $\displaystyle f$ on $\displaystyle I$ based at $\displaystyle a$. Then $\displaystyle F_a$ is uniformly continuous on $\displaystyle I$. If $\displaystyle f$ happens to be continuous on $\displaystyle I$, then $\displaystyle F_a$ is differentiable on $\displaystyle I$ and $\displaystyle F_{a}'(x) = f(x)$ for all $\displaystyle x \in I$.

Proof. Suppose $\displaystyle a \in \mathbb{R}$, $\displaystyle I$ an interval containing $\displaystyle a$, and let $\displaystyle f: I \to \mathbb{R}$ be a function. The area accumulation function $\displaystyle F_a: I \to \mathbb{R}$ is defined by $\displaystyle F_{a}(x) = \int_{a}^{x} f(t) \ dt$. (1) We want to show that $\displaystyle F_a$ is uniformly continuous on $\displaystyle I$. To do this, would we use a sequence approach? Because $\displaystyle F_a$ is not Lipschitz. (2) Suppose $\displaystyle f$ is continuous on $\displaystyle I$. We want to show that $\displaystyle F_a$ is differentiable at some point $\displaystyle c$. So we look at the following:

$\displaystyle \frac{F_{a}(x)-F_{a}(c)}{x-c} = \frac{\int_{a}^{x} f(t) \ dt- \int_{a}^{c} f(t) \ dt}{x-c} = \frac{1}{x-c} \int_{x}^{c} f(t) \ dt$

We want to show that $\displaystyle \lim\limits_{x \to c} \frac{1}{x-c} \int_{c}^{x} f(t) \ dt = f(c)$. And so $\displaystyle f(c) = \frac{1}{x-c} \int_{c}^{x} f(c) \ dt$. The result follows. $\displaystyle \diamond$

Is the second part correct? In the first part, would the easiest way be to use a sequence approach? Or just the regular $\displaystyle \epsilon-\delta$ definition?

2. Originally Posted by manjohn12
Theorem. Let $\displaystyle a \in \mathbb{R}$, and let $\displaystyle I$ be an interval containing $\displaystyle a$. Let $\displaystyle I \subseteq K \subseteq \mathbb{R}$. Suppose that the function $\displaystyle f: K \to \mathbb{R}$ is integrable on $\displaystyle [a,x]$ for all $\displaystyle x \in I$ with $\displaystyle x > a$ and on $\displaystyle [x,a]$ for all $\displaystyle x \in I$ with $\displaystyle x < a$. Let $\displaystyle F_a$ be the area accumulation function for $\displaystyle f$ on $\displaystyle I$ based at $\displaystyle a$. Then $\displaystyle F_a$ is uniformly continuous on $\displaystyle I$. If $\displaystyle f$ happens to be continuous on $\displaystyle I$, then $\displaystyle F_a$ is differentiable on $\displaystyle I$ and $\displaystyle F_{a}'(x) = f(x)$ for all $\displaystyle x \in I$.

Proof. Suppose $\displaystyle a \in \mathbb{R}$, $\displaystyle I$ an interval containing $\displaystyle a$, and let $\displaystyle f: I \to \mathbb{R}$ be a function. The area accumulation function $\displaystyle F_a: I \to \mathbb{R}$ is defined by $\displaystyle F_{a}(x) = \int_{a}^{x} f(t) \ dt$. (1) We want to show that $\displaystyle F_a$ is uniformly continuous on $\displaystyle I$. To do this, would we use a sequence approach? Because $\displaystyle F_a$ is not Lipschitz. (2) Suppose $\displaystyle f$ is continuous on $\displaystyle I$. We want to show that $\displaystyle F_a$ is differentiable at some point $\displaystyle c$. So we look at the following:

$\displaystyle \frac{F_{a}(x)-F_{a}(c)}{x-c} = \frac{\int_{a}^{x} f(t) \ dt- \int_{a}^{c} f(t) \ dt}{x-c} = \frac{1}{x-c} \int_{x}^{c} f(t) \ dt$

We want to show that $\displaystyle \lim\limits_{x \to c} \frac{1}{x-c} \int_{c}^{x} f(t) \ dt = f(c)$. And so $\displaystyle f(c) = \frac{1}{x-c} \int_{c}^{x} f(c) \ dt$. The result follows. $\displaystyle \diamond$

Is the second part correct? In the first part, would the easiest way be to use a sequence approach? Or just the regular $\displaystyle \epsilon-\delta$ definition?

So for (1), would a sequence approach be the best?