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For a topological space $Y$ and topological spaces $X_i$, a function $f:Y\to {\prod }_i{X}_i$ is continuous into the product topology exactly when every coordinate function $f_i={\pi }_i\circ f$ is continuous.^[ref]

Proof outline: one direction uses function composition with the continuous projection maps. For the other, the preimage of a basic cylinder set isa finite set intersection of open sets in $Y$. This gives continuity on every basic test. It explains, for example, why $t\mapsto (t,t^2,t^3,\dots )$ from the real line to ${\mathbb{R}}^{\mathbb{N}}$ is continuous in the product topology: each coordinate is a polynomial function. The general coordinate criterion fails for the box topology, where infinitely many preimages can be required at once. For example, $g(t)=(t,t,\dots )$ has continuous coordinate functions, but the preimage of the box ${\prod }_{j\ge 1}(-1/j,1/j)$ is $\{0\}$, which is not an open set in the real line.

A continuous real function satisfying $g(x+y)=g(x)g(y)$ is either identically zero or $g(x)=e^{cx}$ for a real constant $c$. If it is nonzero somewhere, $g(0)=1$ and $g(x)g(-x)=1$, so it never vanishes. The identity $g(x)=g(x/2{)}^2$ then makes it everywhere positive. Its real logarithm is a continuous additive function, hence linear by additive function continuous at one point is linear. Exponentiating proves the classification, and each listed function satisfies the equation.

A continuous function sends sufficiently nearby function inputs to nearby function outputs. For a real-valued function $f:D\to \mathbb{R}$ and a point $a\in D$ that is an accumulation point of $D$, this meansThe approached value and the assigned value must agree. Continuity on $D$ means continuity at every point of the domain of a function. At an endpoint of an interval, only approaches inside that domain of a function count. The epsilon-delta definition of continuity below also applies at an isolated point, where continuity holds automatically.^[ref]

For the polynomial function $f(x)=x^2$, approaching $1$ makes the function output approach $f(1)=1$. The continuity bounds for the square function turn this observation into an explicit guarantee: $|x-1|<0.2$ implies $|x^2-1|<0.6$. This is useful whenever a small error in an function input must produce a controlled error in an function output. The required input tolerance can depend on the point and on the chosen output tolerance.

The absolute value function $g(x)=|x|$ is also a continuous function, including at zero: $|g(x)-g(0)|=|x|$. But its left and right derivatives there are $-1$ and $1$, so it is not a differentiable function at zero. Continuity controls values without requiring a tangent line. Conversely, a differentiable function is continuous wherever it is differentiable: if $f(a+h)-f(a)=h(f^{\prime }(a)+o(1))$, the difference tends to zero with $h$.

Several useful consequences reuse this one local idea. Addition, multiplication and function composition preserve continuity; a quotient of functions is continuous where its denominator is nonzero. On an interval, the intermediate value theorem turns continuity into a guarantee that no intermediate function output is skipped. On a compact set, the extreme value theorem guarantees attained maximum and minimum values. Those extra hypotheses matter: continuity alone on an unbounded domain of a function does not provide a global bound or an attained global maximum.

A rational function such as $x\mapsto 1/x$ is continuous on its domain of a function $\mathbb{R}\setminus \{0\}$. Zero is excluded from that domain of a function; the missing point prevents a continuous extension there, but it does not make the existing map discontinuous at one of its actual inputs. Always state the domain of a function before judging continuity.

A map $f:X\to Y$ between topological spaces is continuous when every open set $V\subseteq Y$ has an inverse image $f^{-1}(V)=\{x\in X:f(x)\in V\}$ that is an open set in $X$. The notation denotes a subset, so $f$ need not have an inverse function.

For the continuous map $f:\mathbb{R}\to \mathbb{R}$, $f(x)=x^2$, with the usual metric topology,The two open intervals form an open set; an inverse image can split into separate pieces without violating continuity. The defining condition concerns inverse images, rather than images: here $f((-1,1))=[0,1)$ is not an open set in the real line. Checking this one inverse image illustrates the definition; it alone does not prove continuity for all target open sets.

If $X$ is an infinite set with its cofinite topology and $Y$ is a Hausdorff space, every continuous map $f:X\to Y$ is a constant function. Otherwise two distinct image points would have disjoint open neighbourhoods. Their inverse images would be nonempty disjoint open sets of $X$, contradicting its hyperconnected space property.

For example, $f(n)=n$ from ${\mathbb{N}}_{>0}$ with its cofinite topology into the real line with its usual Euclidean metric is not continuous: the inverse image of the open interval $(1/2,3/2)$ is $\{1\}$, which is not open in the infinite domain. A continuous real-valued readout on that domain can only report one value. Finite cofinite domains are different: their discrete topology allows arbitrary functions into any topological space.

A contour plot displays several level curves of a real-valued function of two variables, usually with labels indicating their values. It represents a surface's height on a flat page. Closely spaced level curves indicate a large change per unit distance only when the gaps between their labeled values are comparable.

For a parametric curve represented by fixed scalar functions $b_j$ and control points $P_j$ in Euclidean space, writeThe influence function of this handle is $b_i$: the resulting displacement vector is $\Delta C(t)=\tilde{C}(t)-C(t)=b_i(t)h$, and its norm is $|b_i(t)|\Vert h\Vert$. These are exact finite changes, not only a first-order estimate. Where $b_i$ is negative the displacement points opposite to $h$. Where $b_i(t)=0$, the curve is unchanged at that parameter.

For the cubic Bézier curve above, the Bernstein basis influence function is $b_{1,3}(t)=3t(1-t{)}^2$. Its derivative isIt increases up to $t=1/3$ and decreases thereafter on $[0,1]$. Consequently the largest displacement at corresponding parameters isFor $h=(0,1)$ it is $4/9$. This is a comparison at the same parameter. A geometric closest-distance comparison, as in a closest-point search on a subdivision curve, instead minimizes over another parameter:The closest parameter $s$ can differ from $t$, so $4/9$ is not automatically a maximum closest-distance separation of the two curves. Neither parameter is assumed to be arc length.

The support of a function $b_i$ is the closure of the parameter values where $b_i\ne 0$. It bounds where a nonzero handle move can act, but may include boundary points where the actual displacement is zero. Here the support of a function $b_{1,3}$ is all of $[0,1]$ relative to the parameter domain, while the endpoints are unaffected. In contrast, a B-spline of order $k$ has support within $[t_i,t_{i+k}]$ in its spline knot sequence, so its handle affects only that parameter range; the support and Gram bandwidth of B-splines describes the same locality in another application. Changing the spline knots or the basis along with the handle requires a different comparison.

When the basis functions form a partition of unity, ${\sum }_j{b}_j(t)=1$, each evaluation is an affine combination of the control points. Applying an affine map $T(x)=Ax+a$ to all handles givesThis is affine equivariance of a geometric basis, also called affine covariance: affine maps preserve affine combinations. In particular, translating every handle by $h$ translates the whole curve by $h$, whereas moving just one handle produces $b_i(t)h$. The Bernstein basis has this partition of unity property.^[ref] Nonnegative weights further make evaluations convex combinations, giving the convex hull bound illustrated by the control polygon; nonnegativity is not needed for affine covariance.

For a proof outline of the influence formula, subtract the two basis representations: every unchanged coefficient cancels, leaving $b_i(t)(P_i+h)-b_i(t)P_i=b_i(t)h$. The same subtraction with several moved handles gives $\Delta C(t)={\sum }_i{b}_i(t)h_i$. No claim about interpolation, closest-point search on a subdivision curve or endpoint motion follows until the actual basis values are examined.

For arbitrary topological spaces $X_i$, an index set $I$, a sequence $x^{(n)}\in {\prod }_i{X}_i$, and a point $x$ in that Cartesian product, convergence in a product space to $x$ in the product topology is equivalent to coordinatewise convergence in every factor. No countability or Hausdorff assumption on the factors is needed. For a product space of copies of the real line, regarding each point as a function $I\to \mathbb{R}$ identifies this with pointwise convergence.

Proof outline: the forward implication follows because each projection map is continuous. Conversely, given a basic neighborhood $B(F,U)$ of $x$, each of its finitely many coordinate tests holds for all $n\ge N_i$. Taking $N={\max }_{i\in F}{N}_i$ makes all tests hold simultaneously; if $F$ is empty, any threshold works. This proves eventual membership in every neighborhood, rather than only in a sampled collection of cylinders. The same equivalence holds for nets: a directed set supplies a common upper bound for finitely many thresholds. Sequences alone need not characterize the whole topology of an arbitrary product space.

The convex conjugate records a function using slopes rather than input positions. Let $f:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ be a proper extended-real function: it is finite somewhere and never takes $-\infty$. Its convex conjugate iswhere $p\cdot x$ is the dot product and $\mathrm{dom}f=\{x:f(x)<+\infty \}$ is the effective domain. An infinite value of $f$ excludes that input from the supremum. The value $f^{\ast }(p)$ can be $+\infty$, but cannot be $-\infty$ because an input with finite $f(x)$ supplies a finite lower bound.

The convex conjugate is automatically a convex function, even when $f$ is not convex: each fixed input gives an affine function $p\mapsto p\cdot x-f(x)$, and their supremum is convex. It is also lower semicontinuous. It need not be proper: for $f(x)=-x^2$ on $\mathbb{R}$, $px-f(x)=px+x^2$ is unbounded above for every $p$. The definition and the automatic convexity are discussed in Boyd and Vandenberghe, Convex Optimization (2004), §3.3.1.

The best lower bound with a prescribed slope. Whenever $f^{\ast }(p)$ is finite, the affine functionis an affine minorant of $f$. Indeed, an affine function $p\cdot x+b$ lies below $f$ everywhere exactly when $b\le {\inf }_x(f(x)-p\cdot x)=-f^{\ast }(p)$. Thus $-f^{\ast }(p)$ is the largest possible intercept at slope $p$. The convex conjugate trades positions for slopes: $x$ specifies where to inspect $f$, whereas $p$ specifies which global affine minorant to use. Taking the supremum over all these affine minorants produces the biconjugate; the Fenchel-Moreau theorem says this recovers a proper convex function when it is lower semicontinuous.

A quadratic, computed by completing the square. For the convex quadratic function $f(x)=x^2/2$,Consequently $f^{\ast }(p)=p^2/2$, and the unique maximizing input is $x=p$. The affine minorant ${\ell }_p(x)=px-p^2/2$ is the tangent line at $(p,p^2/2)$: its slope is $p$ and its intercept is $-f^{\ast }(p)=-p^2/2$. For example, $p=2$ gives $f^{\ast }(2)=2$ and the tangent line $2x-2$, touching at $(2,2)$. This is the same curve described either by its point values or by the intercepts of its tangent lines.

A corner turns into a constraint on slopes. For the absolute value $f(x)=|x|$, if $|p|\le 1$ then $px-|x|\le 0$, with value zero at $x=0$. If $p>1$, taking $x\to +\infty$ makes $(p-1)x\to +\infty$; if $p<-1$, taking $x\to -\infty$ does the same. Therefore its convex conjugate is zero on $[-1,1]$ and $+\infty$ outside: it is the indicator functional of a constraint set for this interval. For $|p|<1$ the unique maximizing input is zero; for $p=1$ every $x\ge 0$ maximizes, and for $p=-1$ every $x\le 0$ maximizes. The corner's many supporting slopes are exactly the subdifferential $\partial f(0)=[-1,1]$.

A supremum need not be attained. For a differentiable convex function finite on all of ${\mathbb{R}}^n$, an input maximizes $p\cdot x-f(x)$ exactly when $p=\nabla f(x)$. If $f$ is strictly convex, there is at most one such input; existence is a separate issue. For the exponential function $f(x)=e^x$, the derivative is positive. At $p>0$, solving $p=e^x$ gives $x=\log p$ and $f^{\ast }(p)=p\log p-p$. At $p=0$, the supremum of $-e^x$ is zero as $x\to -\infty$, with no maximizing input. At $p<0$, $px-e^x\to +\infty$ as $x\to -\infty$. Thus a strictly convex differentiable function can have both unattained finite conjugate values and infinite conjugate values.

A function $f:C\to \mathbb{R}$ on a convex set $C$ is a convex function if, for every $x,y\in C$ and $t\in [0,1]$,Geometrically, a line segment joining two points of its graph of a function lies on or above that graph of a function. The convex set condition ensures that the intermediate input stays in the domain of a function. An affine function satisfies the defining inequality with equality.

For the quadratic function $f(x)=x^2$, the inputs $1$ and $3$ have equally weighted arithmetic mean $2$. Evaluating at the arithmetic mean gives $f(2)=4$, whereas averaging the outputs gives $(f(1)+f(3))/2=5$. Averaging first saves one unit in this example. More generally, direct expansion givesThis identity proves convexity and also shows exactly when its defining inequality is strict.

For a differentiable convex function, the tangent line in one dimension, or the corresponding supporting affine function in several dimensions, lies below the entire graph of a function. This is the subgradient inequality. The equivalent geometric statement is that the epigraph, consisting of points on or above the graph of a function, is a convex set. These characterizations are developed in Boyd and Vandenberghe's convex-function slides, sections 3.1–3.4.

Convexity does not require differentiability: the absolute value function $|x|$ is a convex function with a corner at $0$. When a function is twice continuously differentiable on an open interval, it is a convex function exactly when its second derivative is nonnegative throughout that interval. In several dimensions, the corresponding criterion on an open convex set is that its Hessian matrix is positive semidefinite everywhere. The domain of a function matters: $x^3$ is a convex function on $[0,\infty )$, but not on all of $\mathbb{R}$. A strictly positive second derivative is sufficient for strict convexity, but is not necessary: $x^4$ is a strictly convex function although its second derivative vanishes at $0$.

The usefulness of convexity is that local information can certify a global conclusion. Every local minimum of a convex function on a convex set is a global minimum: if a lower value existed elsewhere, the defining inequality would produce lower values arbitrarily close to the proposed local minimum. For a differentiable convex function on an open convex set, a zero gradient therefore certifies a global minimum. Strict convexity gives uniqueness when a global minimum exists; it does not supply existence by itself. These facts explain the role of convex functions in convex optimization.

Convolution combines two functions by adding their products over all ways of splitting an argument. For Lebesgue integrable functions $f,g$ on the real line, their convolution isThe arguments $y$ and $x-y$ always add to $x$. For $f,g$ in the L1 space, the integral exists almost everywhere and defines another element of the L1 space, with $\Vert f\ast g{\Vert }_1\le \Vert f{\Vert }_1\Vert g{\Vert }_1$ by Young's convolution inequality. Particular examples below have a value at every $x$.

To see what the integral counts, take the indicator function $q=1_{[0,1]}$. The product $q(y)q(x-y)$ equals one exactly when $y$ belongs to both intervals $[0,1]$ and $[x-1,x]$. Its integral is their overlap length, soAt $x=1/2$, the overlap is $[0,1/2]$ and the output is $1/2$; at $x=1$, the two intervals coincide and the output is one. The result is the shifted triangular function $\mathrm{tri}(x-1)$. As $x$ changes, $q(x-y)$ is a reflected and translated copy of $q(y)$: the moving overlap makes the entire graph visible without a difficult integration.

The same calculation gives the probability distribution of a sum of two independent random variables. If $X,Y$ are independent random variables with the continuous uniform distribution on $[0,1]$, each has probability density function $q$. Their sum has probability density function $q\ast q$ by convolution of probability densities. The probability density function at one is larger than near zero or two because there are more combinations of the two inputs giving that sum. Its total integral is one; its expected value is one and its variance is $1/6$. These follow equally from integrating the triangular function or from linearity of expectation and variance additivity for independent random variables.

Convolution also gives a useful averaging operation. For $\epsilon >0$, let $k_{\epsilon }(y)=1_{[-\epsilon ,\epsilon ]}(y)/(2\epsilon )$. Thenthe mean of $f$ in a neighborhood of $x$. A smaller $\epsilon$ examines a smaller neighborhood. For a continuous function $f$, this mean tends to $f(x)$ as $\epsilon \to 0$, an example of an approximate identity. Integrating a sharp input against a distributed convolution kernel can produce a smoother output; the triangular example turns two discontinuous inputs into a continuous output.

In signal processing, a linear time-invariant system combines shifted copies of its impulse response by the same rule. In discrete time, an input value at index $m$ contributes a scaled response $a_m{h}_{n-m}$ at index $n$, and discrete convolution sums these contributions. This is the superposition interpretation developed in Oppenheim, MIT Signals and Systems, Lecture 4, convolution sums and integrals. The convolution theorem then gives another description: transform both inputs with a Fourier transform and multiply their transforms. Combining shifts, combining independent quantities and multiplying frequency descriptions are three faces of the same operation.

A convolution kernel is the fixed function $h$ in an operator $T_{h}f=f\ast h$. It specifies how shifted input values are weighted. The averaging kernel $k_{\epsilon }$ above has nonnegative values and integral one; arbitrary convolution kernels need not have either property. A linear time-invariant system uses its impulse response as a kernel when its output admits this representation.

For functions $f,g$ on $[0,\infty )$ with $e^{-\sigma t}f(t)$ and $e^{-\sigma t}g(t)$ Lebesgue integrable functions, their unilateral convolution, defined almost everywhere if necessary, satisfiesTaking a causal extension by zero of both functions gives the existing whole-line convolution definition. Absolute convergence and Fubini's theorem justify the product rule: put $t=\tau +u$ in the double integral, so $e^{-st}=e^{-s\tau }{e}^{-su}$. For piecewise continuous functions of exponential order $a$ and $b$, the hypotheses hold for $\mathrm{Re}s>\max (a,b)$. The Laplace transform article works out the concrete example $h(t)=e^{-2t}$ and the Heaviside step function input, whose output is $(1-e^{-2t})/2$. ^[ref].

Conway's Game of Life is a cellular automaton in which a simple local rule produces patterns that remain still, pulse, or travel across a square grid. Choose the starting pattern; after that there are no player decisions. The Life blinker and Life glider below make the surprise concrete: the rule mentions only births and survival, yet repeated applications create a clock and a moving shape.

The standard board is infinite, with cells at ordered pairs $(x,y)\in {\mathbb{Z}}^2$ of integers. At generation $t\in \{0,1,2,\dots \}$ let $a_t(x,y)$ be $1$ for a live cell and $0$ for a dead cell. Count all eight cells sharing an edge or a corner, excluding the center itself:The local Boolean function defining the next generation isThis is the rule B3/S23: a dead cell is born with exactly three live neighbors; a live cell survives with exactly two or three. A live cell with zero or one, or with four through eight, dies. A dead cell with two stays dead. All counts and center states come from generation $t$: this is a synchronous update, even if an algorithm computes its cells one at a time.^[ref]

In particular, the two births in the Life blinker do not help any other cell survive until the following generation. Writing each answer over its old value and then reading that new value for another count changes the rule. The simplest correct algorithm reads one array and writes the next generation into a second array; swap their roles after the entire step. The one-row in-place Life update achieves the same rule with less auxiliary storage by preserving the old values that later counts still need.

A finite implementation must specify its boundary convention. Holding all cells outside the rectangle permanently dead can destroy a Life glider when it reaches the edge. Wrapping neighbor coordinates from one edge to the opposite edge allows interaction with patterns on the other side. Neither convention is the infinite board. The figures display finite windows into an infinite board initially dead everywhere except the stated live cells; the windows do not impose walls.

Conway's Game of Life illustrates emergence in a precise, modest sense: the Life glider's motion is a collective pattern derived from local birth and death, without a motion command. It has deterministic evolution; visually surprising behavior introduces no random choices. Nor do the words "alive" and "dead" make it an experimentally validated model of biology. The beauty is that the same rule yields a stable four-cell square, the Life blinker, and the Life glider: three qualitatively different behaviors from a handful of cells.

For the stable square, take the live set $\{(0,0),(1,0),(0,1),(1,1)\}$. Each live cell has three live neighbors, and every dead cell has at most two. The next generation is identical: this configuration is a fixed point of the global update function. In contrast, the Life blinker returns after two generations, and the Life glider returns only after allowing a translation.

The patterns are useful as small, exact checks of an algorithm: the square checks survival without births, the Life blinker checks births and deaths in one simultaneous step, and the Life glider checks several steps and diagonal neighbors. More elaborate constructions can encode data and assemble Boolean circuits; Johnston and Greene develop constructions that simulate arbitrary computation in their chapter 9. This is a statement about suitably designed patterns with enough space and time, not a claim that every starting pattern computes something useful or that a fixed finite board has unlimited computer memory.^[ref]

Martin Gardner introduced Conway's Game of Life to his Scientific American readers in October 1970, reporting that Conway selected the rules after experimentation with patterns. The aim was sustained, varied behavior rather than immediate extinction or trivially demonstrable unlimited growth. These were mathematical explorations of chosen rules.^[ref]

Gardner's hands-on procedure is still a useful experiment: place black counters for the current live cells; mark doomed counters by stacking another black counter; mark births with white counters; finish checking before removing the doomed stacks and turning white counters black. Count only the original black occupants. Repeat with three adjacent counters to obtain the Life blinker, then with the five coordinates of the Life glider. Recording each generation tests whether the procedure implements the rule; it does not measure a natural population.^[ref]

The $k$th correlation function is the density of the $k$th factorial moment measure: integrating a nonnegative test function against $R_k$ gives the expectation of its sum over ordered $k$-tuples of distinct points. For exactly $m$ exchangeably labelled points with symmetric joint density $f_m$, it is $m!/(m-k)!$ times the marginal density of $k$ labels. It is not normalized to have total mass one.