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

This strengthens the regularity known for a monotonic function beyond the restriction on its jump discontinuities.

A real-valued function $f:D\to \mathbb{R}$ is a strictly increasing function if every pair $x<y$ in its domain of a function satisfies $f(x)<f(y)$. It is therefore injective, and each output has at most one preimage. Its inverse function on its image of a function is also strictly increasing. Every strictly increasing function is a nondecreasing function; the converse fails when outputs repeat.

The polynomial $f(x)=x^3$ is a strictly increasing function on the real line, although its derivative $f^{\prime }(x)=3x^2$ is zero at zero. Indeed, for $x<y$,because $x^2+xy+y^2=(x+y/2{)}^2+3y^2/4$ can vanish only when $x=y=0$. Thus a positive derivative everywhere is sufficient on a real interval, but is not necessary for a strictly increasing function.

Strict monotonicity does not ensure continuity: $g(x)=x$ for $x<0$ and $g(x)=x+1$ for $x\ge 0$ is a strictly increasing function with a jump discontinuity. Its image of a function omits $[0,1)$, so the inverse function exists only on the actual image of a function, rather than on all the real numbers.

If continuation around two square-root branch points acts on a primitive as $R_A(W)=2A-W$ and $R_B(W)=2B-W$, thenThus pairs of branch-point loops generate additive periods of an inverse function.

The cumulant-generating function is ${\psi }_X(\theta )=\log \mathbb{E}{e}^{\theta X}$, interpreted as an extended-real convex function where the exponential moment may be infinite.

An exponential moment of a random variable $X$ is an expectation $\mathbb{E}[e^{\theta X}]$ or $\mathbb{E}[e^{\theta |X|}]$ at a specified parameter $\theta$. Its finiteness controls tail decay and the domain of the moment-generating function.

The cubic moment curve is ${\mathcal{M}}^3=\{(t,t^2,t^3):0\le t\le 1\}\subset {\mathbb{R}}^3$.

A submodule $N$ of a module $M$ is a subset closed under addition, additive inverses, and scalar multiplication by every element of $R$.

A pregeometry is a finitary closure structure with the exchange axiom, extending the closure description of a matroid to possibly infinite underlying sets. A pregeometry is a closure operator with finite character and exchange. Algebraic closure in a strongly minimal theory is a pregeometry.

The Möbius transformations commuting with $J(z)=1/\overline{z}$ are exactlyThey preserve the open unit disc exactly when $|a|>|b|$. The identityproves the latter assertion.

For determinant-one real representatives, trace magnitude below, equal to, or above two gives elliptic, parabolic, or hyperbolic type.

A condition $\arg ((z-z_1)/(z-z_2))=\theta$ fixes the oriented angle subtended by $z_1,z_2$. Its locus is an arc of a circle through those two points, with the endpoints excluded.

A fixed point of a Möbius transformation satisfies a quadratic equation on the Riemann sphere.

The transformations that take every Euclidean circle to a Euclidean circle are preciselyThey form the stabilizer of infinity in the Möbius group and are not a normal subgroup.

A generalized circle is either a Euclidean circle or a straight line together with the point at infinity. Every Möbius transformation maps generalized circles to generalized circles. A Euclidean circle becomes a line exactly when it contains the pole, the preimage of infinity.

For four distinct points of the Riemann sphere, their cross-ratio iswith the evident limiting conventions at infinity. Every Möbius transformation preserves it.

If adjacent states admit one-step couplings that contract a path metric by a common factor, then the induced Wasserstein distance contracts by the same factor for arbitrary starting distributions.

A sequence of chains exhibits cutoff when its distance from stationarity drops from near one to near zero in a time window negligible compared with its mixing-time scale.

For a finite transitive reversible chain with nontrivial eigenvalues ${\lambda }_i$,

The field has ring of integers $\mathbb{Z}[\sqrt{-33}]$, discriminant $-132$, and Minkowski bound below eight. The nonprincipal ramified ideals above two and three generate four classes, and the split primes above seven introduce no new class. Hence

For $K=\mathbb{Q}(\sqrt[3]{3})$ with ${\mathcal{O}}_K=\mathbb{Z}[\sqrt[3]{3}]$, the field discriminant is $-243$ and the Minkowski bound is $8\sqrt{3}/\pi <5$. The prime ideals above $2$ and $3$ are principal, so every ideal class has a principal representative and