Fundamental Theorem of Calculus, Part II, 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 $\{x\in\mathbb{R}:a<x<b\}$; 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 Riemann integrable on $[a,b]$, and let $F:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable at every point $x\in(a,b)$, with

$$F'(x)=f(x)\qquad\text{for every }x\in(a,b) .$$

Then

$$\int_a^b f(t)\,dt=F(b)-F(a) .$$