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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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