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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 ()x