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If $f\in C^{\infty }(\mathbb{R})$ and every $x$ has some nonnegative integer $n=n(x)$ with $f^{(n)}(x)=0$, then $f$ is a polynomial. The derivative order may vary with $x$; there is no assumed uniform bound.^[ref]

Let $\Omega$ be the open set where $f$ is locally a polynomial. The closed sets $E_n=\{f^{(n)}=0\}$ cover $\mathbb{R}$. Applying the Baire category theorem on every closed interval shows that $\Omega$ is dense. Overlapping polynomials agree, so each connected component of $\Omega$ carries one polynomial.

If $X=\mathbb{R}\setminus \Omega$ were nonempty, it would be closed with no isolated points: adjacent polynomials at an isolated point have matching derivatives and glue across it. Apply the Baire category theorem on $X$ to find an open interval $I$ with $\varnothing \ne X\cap I\subset E_n$. Taking derivatives along accumulation points gives $f^{(m)}=0$ on $X\cap I$ for all $m\ge n$. Every intervening polynomial component has an endpoint there; matching its derivatives forces its degree below $n$. Hence $f^{(n)}=0$ throughout $I$, contradicting $X\cap I\ne \varnothing$. Thus $\Omega =\mathbb{R}$, and overlapping local polynomials give one global polynomial.

In $C([0,1])$ with the uniform norm, the continuous functions that are monotone on some interval of positive length form a meagre set. Indeed, it is enough to use intervals with rational endpoints. For each such interval, the nondecreasing and nonincreasing functions form closed sets with empty interior: a sufficiently small local triangular perturbation breaks the relevant inequality. The Baire category theorem therefore shows that a dense set of continuous functions is monotone on no nontrivial interval.

A meagre, or first-category, set is a countable union of nowhere dense sets.

A subset $A$ of a topological space is nowhere dense when the interior of its closure is empty. Equivalently, every nonempty open set contains a nonempty open subset disjoint from $A$.

Every nonempty countable complete metric space has an isolated point. Otherwise it is the countable union of its singleton subsets, each of which is nowhere dense, contradicting the Baire category theorem.

If such a space has infinitely many points, it has infinitely many isolated points. Indeed, after deleting any finite collection of isolated points, the remaining set is closed, complete, countable, and nonempty. It therefore has a point isolated in the remaining space; because the deleted set is finite, that point is also isolated in the original space.

Let $S$ be a nonempty closed, convex, symmetric subset of a Banach space and suppose ${\bigcup }_{n\ge 1}nS=X$. Baire's theorem puts a ball $B(x,r)$ inside some $nS$. Symmetry also puts $B(-x,r)$ there, and convexity puts the midpoint of $x+z$ and $-x+z$, namely $z$, in $nS$ whenever $\Vert z\Vert <r$. Hence $B(0,r/n)\subset S$.

A Baire space is a topological space in which every countable intersection of open dense sets is dense, equivalently one in which no nonempty open set is a countable union of nowhere-dense sets. Every compact Hausdorff space and every complete metric space is a Baire space.

Magnetocentrifugal acceleration occurs when matter tied to a rotating inclined magnetic field moves outward under the effective centrifugal force. For a Keplerian disc, cold launching is possible when the field tilts by more than $30^{\circ }$ from the vertical.

An Alfvén surface is where the poloidal flow speed equals the poloidal Alfvén speed. Smooth passage through it imposes a regularity condition relating field-line angular velocity and angular momentum.

The total specific angular momentum transported by matter and magnetic stress isand is constant on each magnetic surface.

Ideal induction in a steady axisymmetric flow makes each magnetic surface rotate with a constant field-line angular velocity $\omega (\psi )$.

When $\rho {\mathbf{u}}_p=k{\mathbf{B}}_p$, the flux-surface function $k$ is the mass flux per unit poloidal magnetic flux.

For ${\mathbf{B}}_p=\nabla \psi \times \nabla \varphi$, surfaces $\psi =\text{constant}$ are poloidal magnetic surfaces and $2\pi \psi$ is magnetic flux up to the choice of axial reference value.

The average treatment effect on the treated is $\mathbb{E}[Y(1)-Y(0)\mid A=1]$, averaging over those who actually receive treatment. Their untreated potential outcomes are counterfactual. This can differ from the average treatment effect when effects vary and treatment uptake changes the population mix. Under conditional exchangeability, consistency in causal inference and adequate untreated overlap, it equals $\mathbb{E}[Y\mid A=1]-\mathbb{E}[\mathbb{E}(Y\mid A=0,X)\mid A=1]$. The second term standardizes untreated outcomes to the treated covariate distribution rather than using the crude untreated mean.

A first-order structure is rigid when its identity map is its only automorphism. Every finite linear order is rigid, whereas $(\mathbb{Z},<)$ is nonrigid because every translation is an order automorphism.

Rigidity here means absence of nonidentity automorphisms of a first-order structure; it does not assert the mechanical rigidity of a rod-and-hinge framework.

A family $(f_g{)}_{g\in G}$ of automorphisms satisfying $f_g{f}_h=f_{gh}$ defines a group action. It is faithful exactly when distinct indices define distinct automorphisms, equivalently when $G$ embeds in the automorphism group.

After adjoining a unary function symbol $f$, injectivity, surjectivity, preservation of each operation, and preservation in both directions of each relation form an axiom scheme whose models interpret $f$ as an automorphism.

For an $\mathfrak{m}$-primary ideal $I$ in a Noetherian local ring, $\mathrm{length}(R/I^n)$ agrees for all sufficiently large $n$ with the Hilbert-Samuel polynomial of $I$.

For a Noetherian local ring $(A,\mathfrak{m})$, the pole dimension $d({\mathrm{gr}}_{\mathfrak{m}}(A))$ is the order of the pole at $t=1$ ofThe Hilbert-Serre theorem makes this series rational, so the order is defined.

A constant added to a function changes its reference level without changing its derivative:Geometrically, it is a vertical translation of the graph. For $F(x)=x^3/3$, changing $C$ moves every plotted value by the same amount while preserving every tangent slope. This is the geometric reason that antiderivatives form a family.

When $F$ represents a physical quantity, the physical dimension of $C$ must match that of $F$. When position is obtained from velocity, $C$ is a position offset. An unknown starting position remains unknown even when the rate of change is known exactly.

When describing the family of antiderivatives on an interval, the arbitrary additive constant is the constant of integration. On a disconnected open domain, the corresponding freedom is a locally constant function, as in antiderivatives on disconnected domains, rather than necessarily one constant on the whole domain.