sin x

cos x

100 continuous function

0

0

x2– 1

x– 1

Unraveling the continued-fraction expansions of other

irrational numbers can yield analogous discoveries.

continuous function Informally, a function is said to

be continuous if one can draw its graph without ever

lifting the pencil from the page. This means that the

graph of the function consists of a single curved line

with no gaps, jumps, or holes.

More precisely, a function is continuous at a point

x= a, if the function is defined at the point a, and the

LIMIT

of f(x) as xapproaches aequals the value of the

function at a:

limx→af(x) = f(a)

(If, for example, a function has values f(0.9) = 1.9,

f(0.99) = 1.99, f(0.999) = 1.999, and so on, then one

would be very surprised to learn that f(1) equals 18.

The function would not be deemed continuous at x=

1.) A function that is not continuous at a point is said

to be discontinuous, or to have a discontinuity, at that

point. A function that is continuous at every point in

its domain is called continuous.

Mathematicians have proved that:

i. The sum of two continuous functions is continuous.

ii. The product of two continuous functions is

continuous.

iii. The quotient of two continuous functions is con-

tinuous at each point where the denominator is

not zero.

iv. The

COMPOSITION

of two continuous functions is

continuous.

Since the straight-line graph f(x) = xis continuous, it fol-

lows from properties i and ii that any

POLYNOMIAL

func-

tion p(x) = anxn+…+a1x+ a0is continuous. By property

iii, any

RATIONAL FUNCTION

is continuous at all points

where the denominator is not zero. The functions sin x

and cos xfrom

TRIGONOMETRY

are both continuous.

The tangent function, tan x= , is continuous

at every point other than , the

locations where cosine is zero.

It is possible to remove a discontinuity of a function

at x= aif the limit limx→af(x) exists. For example, the

function f(x) = is not defined at x= 1, since

the quantity has no meaning. Nonetheless, algebra

shows that the limit of this function as xapproaches

the value 1 exists:

(Dividing through by the quantity x– 1 is valid in this

calculation since, for values of xclose to, but not equal

to, 1, the quantity x– 1 is not zero.) Consequently, if

we declare the value of the function to be 2 at x= 1:

we now have a continuous function. A discontinuity

at x= afor a function fis called removable if

limx→af(x) exists.

The issue of continuity is fundamental to the foun-

dation of

CALCULUS

. A study of the

INTERMEDIATE

-

VALUE THEOREM

and its consequences illustrates this.

continuum hypothesis A study of

DENUMERABLE

sets shows that every infinite set contains a denumer-

able subset. Thus, in a well-defined sense, denumerable

sets are the “smallest” types of infinite sets. The

DIAGO

-

NAL ARGUMENT

of the second kind shows that the set

of real numbers is not denumerable, that is, in a mean-

ingful sense, the

CARDINALITY

of the real numbers,

denoted c, is “larger” than the cardinality of denumer-

able sets, which is denoted ℵ0. We have:

ℵ0< c

German mathematician G

EORG

C

ANTOR

(1845–

1918), father of cardinal arithmetic, conjectured that

there is no type of infinite set “larger” than an infinite

set of denumerable objects, but “smaller” than the con-

tinuum of the real numbers. (That is, there is no cardi-

nal number strictly between ℵ0and c.) This conjecture

became known as the continuum hypothesis. It can be

stated equivalently as follows:

Any infinite subset of real numbers can either

be put in one-to-one correspondence with the