CON-STRAIN'ING , p. adj. constraining forces (constraints) . (1) Those forces that tend to prevent a particle's remaining at rest or moving at a uniform velocity in a straight line (accord ing to Newton's first law of motion). (2) Those forces that are exerted perpendicularly to the direction of motion of a particle.
CON-STKUCT ', v. To draw a figure so that it meets certain requirements; usually consists of drawing the figure and proving that it meets the requirements. E.g., to construct a line perpendicular to another line, or to construct a triangle having three given sides.
CON-STRUC'TION , n. (1) The process of drawing a figure ihat will satisfy certain given conditions. See construct. (2) Construc tion in proving a theorem; drawing the figure indicated by the theorem and adding to the figure any additional parts that are needed in the proof. Such "additional" lines, points, etc., are usually called construction lines, points, etc.
CON'TACT , n. chord of contact . See chord.
order of contact . See order- order of contact of two curves.
point of contact . See tangent— tangent to a curve.
CON'TENT, n. content of a set of points. The exterior content (or outer content) of a set of points E is the greatest lower bound of the sums of the lengths of a finite number of intervals (open or closed) such that each point of E Is in one of the intervals, for all such sels of intervals. The interior content (or inner content) is the least upper bound of the sums of the lengths of a finite number of ncinovcr-lapping intervals such that each interval is completely contained in E, for all such sels of intervals; or (equivalently) the difference between the length of an interval I containing E and the exterior content of the complement of E in /. Also called the exterior Jordan content and interior Jordan content. If the exte rior content is equal to the interior con tent, their common value is the (Jordan) content . If the exterior content is zero, then the interior content is also, and the set is said to have (Jordan) content zero . The set of rational numbers in (0, 1) has exterior content 1 and interior content zero; the set 
has content zero. This definition is for sets of points on a line. A similar definition holds for sets in the plane, or in //-dimensional Euclidean space.
CON-TIN'GENCE , n. angle of contingence . The angle between the positive directions of the tangents to a given plane curve at two given points of the curve.
angle of geodesic contingence . For two poinls 
and 
of a curve Con a surface, the angle of geodesic contingence is the angle of intersection of the geodesies tangent to C at 
and 
See above, angle of contingence.
CON-TIN'GEN-CY , n. contingency table (Statistics). If each member of a popula tion can be classified according to each of two criteria and the numbers of categories are p and q, respectively, then there are 
distinct classifications and a table with 
cells showing in each cell the number of members of the population in that classi fication is a 
by 
or 
contingency table. This concept can be extended to cases for which theie are more than two criteria. Following i.s a two-by-three contin gency table with six cells resulting from
classifying a set of 400 persons according to sex and according to opinion on a cer tain political question.
CON-TIN'GENT , adj. contingent annuity and life insurance . See annuity, insur ance— life insurance.
CON-TIN'U-A'TIO N, adj., n. analytic con tinuation of an analytic function of a complex variable, see analytic.
continuation notalion . Three center dots or dashes following a few indicated terms. In case there is an infinite number of terms, the most common usage is to indicate a few terms at the beginning of the set, follow these with three center dots, write the general term, and add three center dots as follows:
continuation of sign in a polynomial. Rep etition of the same algebraic sign before successive terms.
CON-TIN'UED, adj. continued equality . See EQUALITY.
continued fraction . See fraction —con tinued fraction.
continued product . A product of an infinite number of factors, or a product such as 
of more than two factors; denoted by 
that is, capital pi, with appropriate indices. js 
a continued product.
CON'TI-NU'I-TY, n. The property of being continuous. axiom of continuity . See axiom. equation of continuity. A fundamental equation of fluid mechanics, namely, 
where 
is the density of the fluid and 
is the velocity vector. A more general equation takes account of sources and sinks at which fluid is created and destroyed.
principle of continuity. See axiom —axiom of continuity.
CON-TIN'U-OUS, adj. absolutely continuous function. With respect to an interval I , an absolutely continuous function is a function f whose domain contains I and which has the property that, for any positive number 
there is a positive number 
such that if 
is any finite set of non- overlapping intervals such that the sum of the lengths of the intervals is less than n , then 
The definition remains equivalent to this if it is changed io allow a countable number of intervals. An absolutely continuous function is continuous and of b ounded variation, A function / is absolutely continuous on a bounded closed interval 
if and only if there is a function 
such that 
if x is in 
, where the integral may be a Lebesgue integral. Also see RADON — Radon-Nikodym theorem.
continuous annuity. An annuity payable continuously. Such an annuity cannot occur, but has theoretical value. Formulas for this sort of annuity are limiting forms of the formulas for noncontinuous annuities, when the number of payments per year increases without limit while the nominal rate and annual rental remain fixed. The results differ very little from annuities having a very large number of payments per year. Approximate present values for a single life continuous annuity of one dollar is that of a single life annuity payable annually at the end of the year plus 
of a dollar; or that of a single life annuity payable annually at the beginning of the year minus 
of a dollar.
continuous conversion of compound interest. See conversion.
continuous function . A function / whose domain and range arc topological spaces is continuous at a point x if for any neighborhood W of 
there is a neighborhood U of a- such that W coniains all points 
for which u is in U. Such a function f is continuous if it is continuous at each point of D. It then can be proved that f is continuous if and only if t he inverse of each open set in R is open in D (or if and only if the inverse of each closed set in R is closed in D) [see open —open mapping]. For a function f whose domain and range are scis of real or complex numbers, this means that/is continuous at 
can be made as nearly equal to 
as one might wish by restricting x to be sufficiently close to 
; i.e., for any positive number e there is a positive number 
such that 
and x is in the domain of f. This implies that f is continuous at 
if 
is an isolated point of the domain of f (i.e., there is a 
such that there is no x in the domain of/for which 
and 
and that f is continuous at a non-isolated point x 0 if and only if 
A function f is continuous on a set S if it is continuous at each point of S. All polynomial, trigonometric, exponential, and logarithmic functions are continuous at all points of their domains. Allfunctions are continuous at all points at which they are differentiablc. A function 
of two real variables x and y, or of a point 
is continuous at 
if for each positive number e there is a positive number S such that 
if the distance between 
and is 
less than 
and 
is in the domain of/. If 
is nol an isolated point of the domain, this is equivalent to requiring that 
See discontinuity, uniform -uniform continuity.
continuous game . See game.
continuous on the left or right. A real- valued function f is continuous on the right at a point 
if for each 
there is a 
such that 
and to be continuous on the left if for any e > 0 there is a 
such that 
if 
A function is continuous on the right (or left) on an interval (a, b) if it is continuous on the right (or left) at each point of (a, b). See limit —limit on the right.
continuous in the neighborhood of a point. A function is continuous in the neighborhood of a point if there exists a neighborhood of Ihe point such that the function is continuous at each point of the neighborhood. Thus 
is continuous in the neighbor hood of 
if there exists a positive number 
such that 
is continuous at 
for each i , or if
continuous random variable . See RAN DOM —random variable.
continuous surface in a given region . The graph of a continuous function of two vari ables; the locus oi the points whose rectangular coordinates satisfy an equation of the form 
where /is a continuous function of x and y in the region of the 
-plane which is the projection of the surface on lhat plane. E.g., a sphere about the origin is a continuous surface, for 
is a continuous function on, and within, the circle 
To determine the entire sphere, both signs of the radical must be considered. Thought of in this way, the sphere is a multiple (two) valued surface.
continuous transformation . See above, con tinuous correspondence.
piecewise continuous function . See piece- wise.semicontinuous function . If for any arbi trary positive number 
a real-valued funclion f satisfies the relation 
for all x in some neighborhood of 
then f is upper semi continuous at 
; if 
for all x in some neighborhood of 
t hen f is lower semi continuous at 
Equivalent conditions arc, respectively, that the limit superior of 
and that the limit inferior be 
A function is upper scmicontinuous (or lower semicontinuous) on an interval or region R if, and only if, it is so at each point of R. The function defined by 
and 
is upper semicontinuous, but not lower scmicontinuous, at 
CON-TIN'U-UM, n. (pi. eontinud). A compact connected set. It is usually required that the set contain at least two points, which implies thai ii contains an infinite number of points. The set of all real numbers (rational and irrational) is called the continuum of real numbers . Any closed interval of real numbers is a continuum. A continuum is topologically equivalent to a closed interval of real numbers if and only if it does not contain more than two noncut points {see cut).
continuum hypothesis . See HYPOTHESIS —continuum hypothesis.
continuum of real numbers. The totality of rational and irrational real numbers.
CON'TOU'R , adj. contour integral. For a
complex-valued funclion f of complex numbers z and a curve C joining points p and q in the complex plane (or on a Riemarm surface), let 
be 
arbitrary points on C which separate C into n consecutive segments, 
be a point on the closed segment of C which joins 
:o 
and 
be the largest of the numbers Then the cont our integral 
is the limit of 
as 
approaches zero, if this limit exists. If f is con tinuous on C and C is reciifiable, this contour integral exists; if it is also true that F is a func tion such that 
at each point of C, then 
With suitable restrictions on the nature of C, this contour integral can be evaluated as either of the line integrals
where 
is an equation for C and 
with 
and 
and 
real. See cauchy— Cauchy's integral formula, Cauchy's integral theorem.
contour lines . (1) Projections on a plane of all the sections of a surface by planes parallel to this given plane and equidistant apart; (2) lines on a map which pass through points of equal elevation. Useful in showing Ihe rate of ascent of the surface, since the contour lines arc thicker where the surface rises more steeply. Syn. level lines.
CON-TRAC'TJON , n. contraction of a tensor . The operation of putting one eontravariant index equal to a covariant index and then summing with respect to that index. The resultant tensor is called the contracted tensor .
CON'TRA-DIC'TION , n. law of contradic tion. The principle of logic which states that a proposition and its negation cannot both be true; i.e., a proposition can not be both true and false. E.g., for no number x are both the statements 
and 
true.
This is related to the law of the excluded middle, which states that a proposition is true, or its negation is true, or both arc true; i.e., a proposition is true, or it is false, or it is both true and false. E.g., if 
is not true, then 
is true. See dichotomy.
proof by contradiction . Same as reductio AD ABSURDUM PROOF.
CON'TRA-POS'I-TIVE , n. contra positive of an implication. The implication which results from replacing the antecedent by the negation of the consequent and the consequent by the negation of the antecedent. E.g., the contra- positive of "If x is divisible by 4, then x is divisible by 2" is "If x is not divisible by 2, t hen x is not divisible by 4." An implication and its contrapositive arc equivalent —they are either both true or both false. The conira- positive of an implication is the converse of the inverse (or the inverse of the converse) of the implication.