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The intermediate value theorem says that a real-valued continuous function cannot skip a value between two of its outputs. Let $a<b$ and let $f:[a,b]\to \mathbb{R}$ be a continuous function on the entire closed interval $[a,b]$. ThenThe word "between" includes both endpoint values: choose $c=a$ when $y=f(a)$, or $c=b$ when $y=f(b)$. If $y$ is strictly between them, every such $c$ lies in the open interval $(a,b)$. Continuity on $[a,b]$ means continuity at every interior point, together with the endpoint one-sided limits ${\lim }_{x\to a^{+}}f(x)=f(a)$ and ${\lim }_{x\to b^{-}}f(x)=f(b)$. No derivative is required.^[ref]

For finding a zero of a function, the useful special case isStrictly opposite endpoint signs certify a root bracket, because zero is strictly between $f(a)$ and $f(b)$. If an endpoint value is already zero, that endpoint supplies a zero of a function directly; an interior zero of a function need not exist. For example, $f(x)=x$ on $[0,1]$ has its only zero of a function at the left endpoint. The intermediate value theorem guarantees existence, but neither uniqueness nor a necessary sign-change test. Its numerical usefulness is that the bisection method can turn existence into successively tighter root brackets.

A function is locally constant when every point has a neighbourhood on which the function is constant.

For the even convergents of $\sqrt{11}$,Since the second factor has norm one, every such convergent satisfies $p_{2m}^2-11q_{2m}^2=-2$.

If $p_n/q_n$ are the convergents of $\sqrt{11}$, then

If $p_n/q_n=[a_0,\dots ,a_n]$, then appending a positive tail $x$ gives

Every theory containing an inconsistent subtheory is inconsistent, because the same finite derivation of contradiction remains available.

Binary mass transfer is dynamically stable when the donor's radius decreases relative to its Roche-lobe radius after a small mass loss, reducing overflow instead of amplifying it.

Even if both input grammars are regular, the added production $S_0\to S{S}^{\prime }$ has two variables on its right-hand side and therefore need not be a regular production.

Retain every production $\alpha \to \beta$ of $P$ whose right-hand side contains a variable, replace every production $\alpha \to w$ with $w\in {\Sigma }^{\ast }$ by $\alpha \to w{S}^{\prime }$, include $P^{\prime }$, and use $S$ as start variable. If $G$ and $G^{\prime }$ are right-linear, the resulting grammar is right-linear and generates $L(G)L(G^{\prime })$.

A prime knot is a nontrivial knot that is not a composite knot.

The blowup replaces a point of a complex $n$-manifold by the projective space of complex tangent directions ${\mathbb{CP}}^{n-1}$. A biholomorphism carrying one center to another lifts to a biholomorphism of their blowups.

It is the point-center case of blowup of an algebraic variety, also interpreted analytically.

The first Chern class classifies complex line bundles topologically. For a Hermitian holomorphic line bundle with Chern curvature $F_{\nabla }$, Chern-Weil theory gives $c_1(E)=[i{F}_{\nabla }/(2\pi )]$.

The canonical bundle of a complex $n$-manifold is the holomorphic line bundle $K_X={\bigwedge }^n(T^{1,0}X{)}^{\ast }$.

A Kähler manifold is a complex manifold with a positive real closed $(1,1)$-form $\omega$. The associated Riemannian metric is $g(u,v)=\omega (u,Jv)$.

A complex torus is a quotient ${\mathbb{C}}^n/\Lambda$ by a discrete subgroup $\Lambda \cong {\mathbb{Z}}^{2n}$ spanning ${\mathbb{C}}^n$ over $\mathbb{R}$. Translation-invariant differential forms descend to the quotient.

A holomorphic vector bundle is a complex vector bundle whose local trivializations have holomorphic transition functions.

The complexified cotangent bundle decomposes into $(1,0)$ and $(0,1)$ parts. The Dolbeault operator is the type-$(0,1)$ component of the exterior derivative, $\bar{\partial }:{\Omega }^{p,q}\to {\Omega }^{p,q+1}$, and integrability gives ${\bar{\partial }}^2=0$.

Together with its $(1,0)$ counterpart, the Dolbeault operator decomposes the exterior derivative on complex differential forms.

A complex submanifold is locally the common zero set of holomorphic coordinates, equivalently a smooth submanifold whose tangent spaces are complex linear.

The homology object is $H_n=\ker d_n/\mathrm{im}{d}_{n+1}$, formed categorically as the cokernel of the image-to-kernel monomorphism.

A Riemannian manifold is geodesically complete when every maximal geodesic is defined for every real affine parameter, equivalently when every exponential map is defined on its entire tangent space.