Fundamental Theorem of Calculus, Part I, on a Closed Real Interval

Let $a,b$ be real numbers with $a<b$ in the order of the ordered field $\mathbb{R}$, let $[a,b]$ be the closed interval determined by $a$ and $b$, regarded as a subset of the real line $(\mathbb{R},d_{\mathbb{R}})$, and let the codomain $\mathbb{R}$ carry the same metric $d_{\mathbb{R}}$. Write $(a,b)$ for $\{x\in\mathbb{R}:a<x<b\}$ and $(a,b]$ for $\{x\in\mathbb{R}:a<x\le b\}$; every $x\in(a,b)$ is an interior point of the interval $[a,b]$, since $a,b\in[a,b]$ and $a<x<b$. Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$.

Then the following hold.

1. (The indefinite integral is defined) For every $x\in(a,b]$ the restriction of $f$ to $[a,x]$ is Riemann integrable on $[a,x]$. Define $F:[a,b]\to\mathbb{R}$ by

$$F(x)=\int_a^x f(t)\,dt\qquad(x\in[a,b]),$$

where for $x\in(a,b]$ the symbol $\int_a^x f(t)\,dt$ denotes the Riemann integral of the restriction of $f$ to $[a,x]$, and $\int_a^a f(t)\,dt=0$ by the degenerate-interval convention of Mean-Square Riemann Integral of a Family of Random Variables.

2. (Continuity) $F$ is continuous on $[a,b]$.

3. (Differentiability) For every $x\in(a,b)$, $F$ is differentiable at $x$ and

$$F'(x)=f(x) .$$