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# LIMIT AND CONTINUITY OF FUNCTION

By Ralph Ferguson,2014-11-26 12:11
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LIMIT AND CONTINUITY OF FUNCTION

LIMIT AND CONTINUITY OF FUNCTION

1. Function

Definition : If x and y are two variables, then y is said to be a function of x if and only if

the following conditions are both satisfied.

(1) For any x in domain, there exists a corresponding value of y in range.

(2) there is one and only one value of y corresponding to each value of x.

Remark : A function y = f(x) can be denoted by f : A ~ B

127

319

10-7

::

::(relation)

AB

where A is a set called domain which contains all values of x.

where B is a set called range which contains all values of y.

Example 1: Which of the following is/are function(s) mapping from R to R.

(1)

y

x

(2)

y

x

(3)

y

x

page 1

2. Types of function

(1) Odd function : f(x) = f(x) for all real x.

y

x

e.g. f(x) = x, f(x) = sin x, ...

(2) Even function : f(x) = f(x) for all real x.

y

x

2e.g. f(x) = x, f(x) = cos x, ...

(3) Periodic function : f(x + h) = f(x) for all real x, where h is a positive real constant.

yh

h

xh

h

e.g. f(x) = sin x, f(x) = tan x, ...

Remark : (1) h is not unique.

(2) the smallest value of h is called the period.

(4) Injective (one to one) function : For any real x and x, 12

f(x) = f(x) ) x = x . 1212

(5) Surjective (onto) function : For any real y in range, there must exist x in domain 11

such that f(x) = y . 11

Remark : (1) If f(x) is both injective and surjective, then f(x) is said to be bijective.

(2) A function f(x) has inverse function if and only if f(x) is bijective.

(6) Bounded function :

(a) f(x) is said to be bounded from above (upper bounded)

( There exists a constant M such that f(x) M for any real x. 11

(b) f(x) is said to be bounded from below (lower bounded)

( There exists a constant M such that f(x) ? M for any real x. 22

(c) f(x) is said to be bounded

page 2

( There exists a constant M such that M f(x) M for any real x.

Example 2:

4(1) Prove that f(x) = x is even

3(2) Prove that f(x) = x is odd

(3) Prove that f(x) = tan x + sin2x is periodic

3(4) Prove that f(x) = x is injective

(5) Prove that f(x) = 3x 1 is injective

(6) Disprove that f(x) = sin x is not injective

3(7) Prove that f(x) = x is surjective

(8) Prove that f(x) = 3x 1 is surjective

(9) Disprove that f(x) = sin x is not surjective

x(10) Prove that f(x) = e is injective but not surjective (11) Let f be a non-constant function on R such that for any x, y ? R , f(x+y) = f(x)f(y).

i) Show that f(0) = 1. n ii) Show that for all integer n, f(nx) = [f(x)] , where x is a real number.

iii) Show that f is positive for all real values of x.

iv) Hence show that f cannot be bounded.

3. Limit of function

f(x) (1) Right-hand limit :

l fl()xlim;xa~

x a0

If f(x) tends to a value l as x tends to a value a from right-hand direction, then f(x) is

said to have a right-hand limit l on that point a.

(2) Left-hand limit : f(x)

fl()x limlxa~

x0a

If f(x) tends to a value l as x tends to a value a from left-hand direction, then f(x) is

said to have a left-hand limit l on that point a.

Remark : (1) If the right-hand limit equals to the left-hand limit, then the limit of the

function exists, otherwise limit does not exist.

(2) may not be equal to f(a). f()xlimxa~

Consider the following cases:

(a) exists and equals to f(a) f()xlimxa~

2x1e.g. f(x) = and a = 2 x1

in this case, f()x = f(2) = 3 limx~2

(b) fexists but dose not equal to f(a) ()xlimxa~

2x1e.g. f(x) = and a = 1 x1

in this case, f = 2 but f(1) is undefined ()xlimx~1

page 3

(c) does not exist but f(a) is defined f()xlimxa~

e.g. f(x) = [x] (integral part of x) and a = 1

in this case, = 0 but = f(1) = 1 f()xf()xlimlim;x~1x~1

(d) does not exist and f(a) is undefined f()xlimxa~

1e.g. f(x) = and a = 0 x

in this case, and f(0) are both undefined f()xlimx~0

Definition : A function f(x) defined on a set S of real numbers is said to be bounded in S if

there exists a positive number M such that M f(x) M for all x in S.

1e.g. f(x) = is lower bounded by 0 and upper bounded by 1. 2;1x

Definition : A function f(x) is said to be monotonic increasing (decreasing) if and only if

f(a) ? f(b) ( f(a) f(b) ) whenever a > b.

Theorem : If = k and = l, then f()xg()xlimlimxa~xa~

(1) = k ? l fg()()xx?，(limxa~

(2) = kl fg()()xx，(limxa~

f()xk(3) = , provide that l ? 0 limxa~g()xl

sinxTheorem : (1) = 1 limx~0x

1x (2) (1;) = e limx~；x

Example 3: Evaluate

，！！！??;coscoscoscos(1) ?limn??n~；2482

sinxsina(2) limxa~xa

4. Continuous Function

Definition : A function f(x) defined on a set of S of real numbers is said to be continuous

at a point a if and only if = f(a). f()xlimxa~

Remark : (1) A function f is continuous on an interval (C) if its graph contains no breaks or

jumps on the interval. Otherwise, the function is said to be discontinuous (C).

f(x)

C

break

x

Cjump

(2) f= f(a) means both the following conditions are satisfied: ()xlimxa~

i) f(a) is defined

page 4

ii) exists f()xlimxa~

iii) the above two values are equal

Definition : A function is said to be continuous on an interval if it is continuous at every

point of the interval.

Remark : (1) If a is the left-hand end point of a curve f(x), then f(x) is defined to be

continuous at x = a if = f(a). f()xlim;xa~

(2) If b is the right-hand end point of a curve f(x), then f(x) is defined to be

continuous at x = b if = f(b). f()xlimxb~

Definition : A function is said to be discontinuous at a point if it is not continuous at

that point.

Theorem : If f(x) and g(x) are continuous at a, then each of the following functions is

also continuous at a :

(1) f(x) ? g(x)

(2) f(x)g(x)

f(x)(3) provide that g(a) ? 0 g(x)

Corollary : If f(x) and g(x) are continuous on [a, b], then each of the following functions

is continuous on [a, b] :

(1) f(x) ? g(x)

(2) f(x)g(x)

f()x(3) provide that g(x) ? 0 for all x ? [a, b] g()x

Theorem : If u = g(x) is continuous at the point x, i.e. u = g(x), and y = f(u) is continuous ooo

at the point u, i.e. y = f(u), then the composite function y = f(g(x)) is also ooo

continuous at the point x. o

Theorem : If f is a continuous function, then = fg(()x)fg((x))limlimxa~xa~

Remark : The above result is not always true if f is not continuous.

01,whenu?sinxeg. f(u) = and u = g(x) = x11,whenu?

= 0 but = 1 fg(()x)fg((x))limlimx~0x~0

5. Types of discontinuity

(1) Simple discontinuity : The limits of f(x) when x tends to a from either sides exist, but

they are unequal.

00,whenxeg. f(x) = has a simple discontinuity at x = 0 10,whenx??

f(x)

1

x0

page 5

(2) Infinite discontinuity : = for some point a. f()xlimxa~

1eg. f(x) = has an infinite discontinuity at x = 0 x

f(x)

x0

(3) Oscillating discontinuity : At least on one side the function undergoes more and

more oscillations when x tends to a.

1eg. f(x) = sin has an infinite discontinuity at x = 0 x

When x tends to zero, the function f(x) oscillates between 1 and 1.

(x)f

1

x111032

-1

(4) Removable discontinuity : exists, but f(a) is undefined or is different from f()xlimxa~

the approaching value.

2x1eg. f(x) = x1

When x tends to 1, exists but f(1) is undefined. f()xlimx~1

f(x)

x01

6. Intermediate Value Theorem

If f(x) is continuous on [a, b] and f(a) f(b), then there exists c in (a, b) such that f(a) f(c) f(b).

Corollary : If f(x) is continuous on [a, b] and f(a) = f(b) = 0, then there exists c in (a, b)

such that f(c) = 0.

page 6

Example

(1) Show that

page 7

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