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A bounded closed real interval has the form $[a,b]=\{x\in \mathbb{R}:a\le x\le b\}$.

A subset of the real numbers is connected exactly when it is a real interval. If it contains $a<b$ but omits some $c\in (a,b)$, intersecting it with the two open half-lines separated by $c$ gives a separation.

An interval map is a continuous function $F:I\to I$ from a real interval to itself. Its iterates form a one-dimensional discrete dynamical system.

Every real-valued monotone function on a real interval is differentiable with a finite derivative at almost every point.

A unit-speed metric geodesic in a metric space is an isometric embedding $\gamma :I\to X$ of a real interval. Every restricted segment realizes the distance between its endpoints. A one-point interval permits a constant segment. In a unit-edge graph, shortest edge paths parametrized by metric path length are metric geodesics; no smooth structure is required. This is stronger than the differential definition of a geodesic in a Riemannian manifold, which need only minimize locally: on a unit circle, an arc of angle $3\pi /2$ is a Riemannian geodesic but its endpoint distance is only $\pi /2$.

First, equality of two $U_0$ values forces equality of the corresponding $V_0$ values: neither strict comparison holds for $U_0$, and the same is then true for $V_0$ in both directions. Thus $V_0=g\circ U_0$ for a well-defined strictly increasing function $g$ on $J=U_0(\mathcal{P})$. Since $\mathcal{P}$ is a convex set and $U_0,V_0$ are affine functions, $J$ is a real interval andIf $J$ contains $u<v$, let $a=[g(v)-g(u)]/(v-u)>0$ and $b=g(u)-au$. The mixture identity gives $g(w)=aw+b$ for $w\in [u,v]$. For $w>v$, express $v$ as a convex combination of $u$ and $w$ and solve the same identity for $g(w)$; for $w<u$, express $u$ as a mixture of $w$ and $v$. Hence the formula holds on all of $J$, not merely between the chosen anchors. We obtain positive affine uniqueness of affine preference representations:If $U_0$ is constant, the same comparisons force $V_0$ to be constant; choose $a=1$ and the appropriate $b$. If the probability-measure set is empty, the assertion is vacuous. These degenerate cases do not require distinct anchors.

Use the cyclic logarithmic coloringHere the floor function places $x$ in the unique half-open real interval $[(3/2{)}^j,(3/2{)}^{j+1})$. If $y/x\in [1.9,2]$, thenbecause $3/2<1.9\le 2<(3/2{)}^2$. For any real $t$ and $1<u<2$, $\lfloor t+u\rfloor -\lfloor t\rfloor$ is either $1$ or $2$. Applying this to the logarithms of $x$ and $y$ shows that their bin indices differ by $1$ or $2$, and hence have different residues in modular arithmetic modulo $3$. This proves the required separation, including both endpoints of the prescribed ratio interval. The half-open bins remove any ambiguity at their boundaries.

First recall from the least-upper-bound property that every real interval is connected. For completeness, if $[a,b]=U\cup V$ were a separation with $a\in U$, let $s$ be the supremum of the points $t$ for which $[a,t]$ remains in $U$. Relative openness rules out both $s\in U$ and $s\in V$: in the first case one can move slightly to the right while staying in $U$, and in the second one finds points of $V$ slightly to the left of $s$, contrary to the definition of $s$. Thus no separation exists.

Part (a) now makes $g([0,1])$ a connected subset of the real line. Such a set is an interval: if it contained $u<v$ but omitted $y\in (u,v)$, its intersections with $(-\infty ,y)$ and $(y,\infty )$ would separate it. Since $g([0,1])$ contains $g(0)$ and $g(1)$, it consequently contains every number between them. Therefore for each such $y$ there is an $x\in [0,1]$ with $\boxed{g(x)=y}$. This derives the intermediate value theorem from connectedness.

Two real affine functionals on a convex set of lotteries that represent the same strict preferences differ by $V=aU+b$ with $a>0$. Equal utility values must first be shown to give equal values of the other functional. The induced function on the real interval of utility values preserves mixtures; anchor interpolation, including exterior points represented through an anchor, makes it linear throughout. Constant representations are handled separately and still admit a positive slope.

A real tree is a metric space $(T,d)$ in which every pair $a,b\in T$ is joined by a unique arc and that arc is isometric to the real interval $[0,d(a,b)]$. Equivalently, it is a geodesic metric space with no nontrivial simple loops.