Every continuous function on an interval has a primitive; two primitives differ by a constant; and ${\int }_a^{b}f=G(b)-G(a)$ for any primitive $G$

Let $I\subseteq \mathbb{R}$ be order-convex with at least two elements (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $f:I\to \mathbb{R}$ be continuous on $I$ (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point). Call $G:I\to \mathbb{R}$ a primitive of $f$ on $I$ when $G$ is differentiable at every point of $I$ as a function on $I$ with $G^{\mathrm{\prime }}=f$ there (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set). Then:

1. Existence. Fix $c_0\in I$. The function $F:I\to \mathbb{R},F(x):={\int }_{c_0}^{x}f$ is defined at every $x\in I$ (The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$, The integral function $F(x):={\int }_a^{x}f$ of an integrable $f$) and is a primitive of $f$ on $I$.
2. Uniqueness up to a constant. If $G_1$ and $G_2$ are primitives of $f$ on $I$ then there is a real $k$ with $G_1(x)=G_2(x)+k$ for every $x\in I$.
3. Evaluation. If $a,b\in I$ with $a<b$ and $G$ is any primitive of $f$ on $I$, then ${\int }_a^{b}f=G(b)-G(a)\mathrm{.}$

The scope is exactly the continuous case, and that is not a limitation of the proof. An integrable function need not have a primitive, and a function with a primitive need not be integrable; this corollary is precisely the intersection where both fundamental theorems apply, and both witnesses are on the companion page.

• Steps 1.2, 2.1 and 3.1 are the only work beyond citing the two fundamental theorems. The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F^{\mathrm{\prime }}(c)=f(c)$; in particular a continuous $f$ has $F$ as a primitive is stated on a closed bounded interval, while $I$ here may be open, half-open or unbounded, so the derivative it produces is the derivative of a restriction. What those steps supply is a closed subinterval $[u,v]\subseteq I$ that contains all points of $I$ within $\eta$ of $c$, after which the difference quotients of $F$ and of the restriction agree on a punctured neighbourhood and the $\epsilon$-$\delta$ statement transfers verbatim.

• "Two primitives differ by a constant" is not re-minted here. A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant already states exactly that, in its second clause, for functions with equal derivatives on an order-convex domain; claim 2 is that statement applied to $G_1-G_2$.

• Order-convexity of $I$ is essential to claim 2 and harmless elsewhere. On a domain in two pieces a function may be constant on each with different constants, which is why A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant carries the same hypothesis. Claims 1 and 3 use it only to know that closed subintervals spanned by points of $I$ lie in $I$.

• Forward references, orientation only. The two witnesses bounding the scope of this corollary are The sign function is Riemann integrable on $[-1,1]$ and has no primitive there ↗ and A function differentiable on $[0,1]$ whose derivative is unbounded, hence not Riemann integrable ↗ on the companion page; nothing above depends on either.