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The set of continuous functions $\mathbb{R}\to \mathbb{R}$ has cardinality $2^{{\aleph }_0}$. Constant functions provide the lower bound. Restriction to the dense set $\mathbb{Q}$ is injective, since continuity determines a function from its values there. Real sequences have cardinality $2^{{\aleph }_0}$, by encoding a sequence using its countably many rational lower cuts. Restriction therefore supplies the upper bound and Cantor-Schröder-Bernstein gives equality.

A continuous real-valued function is a continuous map whose codomain is the real line.

Continuous real-valued functions on a space with a countable dense subset are determined by their restrictions to that subset. The equality set of two such functions is closed and contains the dense subset. Hence there are at most $|\mathbb{R}{|}^{{\aleph }_0}=2^{{\aleph }_0}$ such functions. If the domain is nonempty, distinct constant functions give the matching lower bound.

For $a\in L$, ambient separation forms $S=\mathcal{P}(a)\cap L$. Ambient replacement bounds the first constructible stages of all members of $S$. At a stage containing $a$ and all of $S$, the latter is definable by the bounded property of being a subset of $a$. Thus $S$ belongs to the next constructible stage and is the internal power set in $L$.

Repeatedly apply a constraint propagation operation that only deletes values from finite constraint domains, preserves their order, and preserves every valid assignment. The sum of domain sizes is a nonnegative loop variant at every changing step, so repetition reaches a fixed point. For $N$ initial candidate values there are at most $N$ changing steps. An empty domain proves failure; a non-singleton fixed point may still require backtracking or explicit enumeration. Preservation of solutions, rather than mere deletion, is essential.

The graph with one vertex per variable and an edge between distinct variables occurring in a common constraint scope. Each hypergraph edge of the constraint hypergraph becomes a clique. A constraint satisfaction problem consisting only of all-different constraints on a common constraint domain is exactly a proper graph colouring problem on this primal graph.

A consistent set of first-order sentences is maximal when adding any new sentence makes it inconsistent. It is then deductively closed and decides every sentence by containing exactly one of it and its negation. Conversely a consistent deductively closed set that decides every sentence is maximal. Merely listing a finite axiom set for a complete theory does not make that raw list maximal: maximality applies to the deductive closure.

A first-order theory is consistent when it has no formal proof of a contradiction. This syntactic property concerns the specified calculus; its equivalence with having a first-order model uses the completeness theorem.

Let $p_1,\dots ,p_n$ be conjugate directions for a symmetric positive-definite matrix $A$. Starting from any $x_1$, minimize $x^{T}Ax/2-b^{T}x$ successively along those full lines, stopping early if the linear-system residual $r_j=b-A{x}_j$ vanishes. The quadratic minimization along a line uses ${\alpha }_j=p_j^T{r}_j/(p_j^{T}A{p}_j)$ and gives $r_{j+1}=r_j-{\alpha }_{j}A{p}_j$.

This makes $p_j^T{r}_{j+1}=0$. For $i<j$, conjugacy preserves $p_i^T{r}_{j+1}=p_i^T{r}_j$. By mathematical induction, $r_{j+1}$ is orthogonal to all preceding search vectors. The $n$ conjugate directions are linearly independent and hence form a basis, so $r_{n+1}=0$. Thus the method reaches the unique solution of the linear system in at most $n$ steps in exact arithmetic. The conjugate gradient method generates such directions without storing a complete prescribed basis in advance.

A congestion window bounds a sender's outstanding data according to congestion control. In TCP it is local sender state, distinct from the advertised receive window.

Fair queueing separates traffic into queues and schedules their service to control sharing, reducing the ability of an unresponsive sender to dominate unrelated traffic merely by filling one common queue.

Assign each vertex of a finite directed graph a label in a total order. To find the least label reachable from each vertex, first compress its strongly connected components. Give each component its own least label, then process the condensation graph in reverse topological order, replacing each label by the minimum of itself and its successors' labels. Every reachable component is either the current one or reached through a first outgoing edge; induction over this order proves the recurrence. The graph-processing work is $O(|V|+|E|)$ with constant-cost label comparisons. Store a representative vertex as well when a witness is required.

Let $N$ have a Poisson distribution with parameter $\lambda$, and let $Y_1,Y_2,\dots$ be independent random variables with a Poisson distribution of common parameter $\theta$, independent also of $N$. Then $Z={\sum }_{i=1}^N{Y}_i$ has a compound Poisson distribution, with $Z\mid N\sim \mathrm{Poisson}(\theta N)$. The law of total expectation yields the displayed probability generating function and expected value $\lambda \theta$. The law of total variance gives variance $\lambda \theta (1+\theta )$, exceeding the mean when both parameters are positive. This extra variation comes from the random number of groups. Jerzy Neyman is the statistician for whom the distribution is named.

If $R\subseteq R^{\prime }$ and $S\subseteq S^{\prime }$, then $R\circ S\subseteq R^{\prime }\circ S^{\prime }$. A witness joining a pair through $S$ and then $R$ also joins it through the larger relations. If both $R$ and $S$ are reflexive relations on the same set, inserting an identity step gives $R\cup S\subseteq R\circ S$, so monotonicity also gives $(R\cup S{)}^2\subseteq (R\circ S{)}^2$.

If $g$ is a Lebesgue measurable function and $h$ is a Borel measurable function, then $h\circ g$ is a Lebesgue measurable function. For an open set $U$, $h^{-1}(U)$ is a Borel set. By measurability of $g$, $g^{-1}(h^{-1}(U))$ is a Lebesgue measurable set, and it equals $(h\circ g{)}^{-1}(U)$. This is composition of measurable functions with the intermediate Borel sigma-algebra stated explicitly. Merely knowing both functions are Lebesgue measurable would not ensure that the intermediate sigma-algebras match.

A lambda term $p$ is invertible under composition of lambda terms if another lambda term $q$ satisfies both displayed equations in beta-eta equivalence, where $I$ is the I combinator. An inverse on just one side does not by itself justify using the inverse equation on the other side.

The complex complex exponential function has nonzero derivative and maps the strip onto the annulus, with fibers given by translations by $2\pi i$. It is locally conformal, but not injective. No biholomorphism from the full strip onto an annulus exists because the strip is simply connected space and the annulus has a loop of nonzero winding number. The complex exponential function is a covering map realizing that topological distinction.

If independent random variables $X_n$ converge almost surely to a finite $X$, then ${\mathrm{limsup}}_n{X}_n$ is measurable for the tail sigma-algebra. The Kolmogorov zero-one law forces every rational threshold event for that limit to have probability zero or one. A real-valued probability distribution with this property is a point mass, so $X$ is almost surely constant. Consequently the Second Borel-Cantelli lemma upgrades almost sure convergence of independent sequences to complete convergence.

In lambda calculus, the closed lambda term $K\equiv \lambda xy.x$ discards its second argument. In combinatory logic, its primitive constant satisfies $KAB=A$. The corresponding principal type in the simply typed lambda calculus is $\alpha \to (\beta \to \alpha )$.