The integral function of a bounded integrable $f$ is Lipschitz, hence uniformly continuous

Let $a<b$ be reals, let $f:[a,b]\to \mathbb{R}$ be integrable (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$), let $K\ge 0$ be a real with $\mathrm{\mid }f(t)\mathrm{\mid }\le K$ for every $t\in [a,b]$ (Lower bound, bounded below, bounded set), and let $F$ be the integral function of $f$ (The integral function $F(x):={\int }_a^{x}f$ of an integrable $f$). Then

$\mid F(y)-F(x)\mid \le K\mathrm{\mid }y-x\mathrm{\mid }\text{for all }x,y\in [a,b],$

that is, $F$ is Lipschitz with constant $K$ on $[a,b]$ (Lipschitz map, $\alpha$-Hölder map for rational $0<\alpha \le 1$, and contraction, Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace). Consequently $F$ is uniformly continuous on $[a,b]$ (Uniform continuity of $f:A\to \mathbb{R}$: one $\delta$ serving every pair of points of $A$) and hence continuous there (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).

No continuity of $f$ is assumed. This is the strongest regularity of $F$ available before the fundamental theorem, and it is what makes the hypotheses of that theorem visible as hypotheses: continuity of $f$ at a point buys differentiability of $F$ there, and integrability alone already buys this much everywhere.

• The estimate is written out on both sides of the diagonal. Hiding the case $y<x$ inside the absolute value would conceal the fact that ${\int }_x^{y}f$ is then the oriented integral of The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$, and that the published inequality $\mid {\int }_p^{q}f\mid \le {\int }_p^q\mathrm{\mid }f\mathrm{\mid }$ is available only for $p\le q$. Step 4.1 is what pays for that.

• The constant is any bound on $\mathrm{\mid }f\mathrm{\mid }$, and it need not be sharp. If $f$ is integrable then it is bounded by definition of the Darboux sums, so some $K$ exists; the theorem is stated with $K$ given rather than existentially, because every later use supplies its own bound.

• The dictionary lemma is cited on purpose. Lipschitz and uniform continuity are defined in this library both for real functions and for maps of metric spaces, and Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace is the single item recording that the two families of notions coincide. Citing it, rather than proving the implication again, is what keeps the library from carrying two notions of continuity.