Title: Functions, Limits and Continuity

Title: Functions, Limits and Continuity

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Title: Functions, Limits and Continuity

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1. Title:Functions, Limits and Continuity Prof. Dr. Nasima Akhter And Md. Masum Murshed Lecturer Department of Mathematics, R.U. 29 July, 2011, Friday Time: 6:00 pm-7:30 pm

2. Outline • Functions and its graphs. • One-one, Onto and inverse functions. • Transcendental functions. • Bounded and monotonic functions. • Limits of functions. • Right and left hand limits. • Special limits. • Continuity. • Right and left hand continuity. • Sectional continuity. • Uniform continuity, Lipschitz continuity.

3. Outline • Functions and its graphs. • One-one, Onto and inverse functions. • Transcendental functions. • Bounded and monotonic functions. • Limits of functions. • Right and left hand limits. • Special limits. • Continuity. • Right and left hand continuity. • Sectional continuity. • Uniform continuity, Lipschitz continuity.

4. Functions and its Graphs X Y f y = f (x) x

5. Functions and its Graphs x f(x)

6. Functions and its Graphs X f Y y = f (x) x f :X → Y if for each x ∊ X ∃ a unique y ∊ Y such that y = f(x).

7. f Functions and its Graphs f f is not a function f is not a function f f f f is a function f is a function f is a function

8. Functions and its Graphs Real valued function X f Y = R f[X] = {f(x) : x ∊ X} Range of f y = f (x) x Domain of f Co-domain of f f :X → Y if for each x ∊ X ∃ a unique y ∊ Y such that y = f(x).

9. Functions and its Graphs Set function X f Y Range of f f[X] = {f(x) : x ∊ X} Class of sets y = f (x) Co-domain of f Domain of f f :X → Y if for each x ∊ X ∃ a unique y ∊ Y such that y = f(x).

10. Functions and its Graphs X f Y Range of f f(A) A f[A] = {f(x) : x ∊ A} y = f (x) Co-domain of f Domain of f If f :X → Y then ∃ two set functions f : 2X→ 2Yand f-1 : 2Y→ 2X

11. Functions and its Graphs X f Y B f-1(B) f -1(B) = {x ∊ X : f(x) ∊ B}. If f :X → Y then ∃ two set functions f : 2X→ 2Yand f-1 : 2Y→ 2X

12. Functions and its Graphs f f ({1, 3, 4}) = {a, b, d}, f ({2, 3}) = {a, d} f ({1, 2}) = {a}, also f -1({a, b, d})= {1, 2, 3, 4}, f -1({a})= {1, 2}, f -1({b, c})= {4}, f -1({c})= Ø,

13. Functions and its Graphs Between these two set functions f -1 plays very important role in topology and measure. Since f -1 preserves countable union, countable intersection, difference, monotonicity, complementation etc. i.e. Theorem If f :X → Y then for any subset A and B of Y, (i) f -1(A ∪ B) = f -1(A) ∪f -1(B). (ii) f -1(A ∩ B) = f -1(A) ∩f -1(B). (iii) f -1(A ∖ B) = f -1(A) ∖f -1(B). (iv) If A ⊆ B then f -1(A) ⊆f -1(B). (v) f -1(Ac) = (f -1(A))c. And, more generally, for any indexed {Ai} of subsets of Y, (vi) f -1( ∪i Ai) = ∪if -1(Ai). (vii) f -1(∩i Ai) = ∩if -1(Ai).

14. Functions and its Graphs X Y x y (x, y) X ⨉ Y = {(x, y) : x ∊ X , y ∊ Y} A relation from X to Y is a subset of X ⨉ Y.

15. Functions and its Graphs A function can also be described as a set of ordered pairs (x, y) such that for any x-value in the set, there is only one y-value. This means that there cannot be any repeated x-values with different y-values. A relation is called a function if for any x-value in the set, there is only one y-value. This means that there cannot be any repeated x-values with different y-values. g f f is not a function g is a function f = {(1, a), (1, b), (2, c), (3, c), (4, d)} is not a function. g = {(1, a), (2, a), (3, d), (4, b)} is a function.

16. Functions and its Graphs X, A subset of R Y, A subset of R y = f(x) x f (x, y) We plot the domain X on the x-axis, and the co-domain Yon the y-axis. Then for each point x in X we plot the point (x, y), where y = f(x). The totality of such points (x, y) is called the graphof the function.

17. Functions and its Graphs The Vertical Line Test This is not the graph of a function. The vertical line we have drawn cuts the graph twice. This is the graph of a function. All possible vertical lines will cut this graph only once.

18. Functions and its Graphs Now we consider some examples of real functions. Example 1. The identity function. Let the function f : R → R be defined by f(x) = x for all real x. This function is often called the identity function on R and it is denoted by 1R. Its domain is the real line, that is, the set of all real numbers. Here x = y for each point (x, y) on the graph of f. The graph is a straight line making equal angles with the coordinates axes (see Figure-1 ). The range of f is the set of all real numbers. Figure-1 Graph of the identity function f(x) = x.

19. Functions and its Graphs Example 2. The absolute-value function. Consider the function which assigns to each real number x the nonnegative number |x|. We define the function y = |x| as From this definition we can graph the function by taking each part separately. The graph of y = |x| is given below. Figure- 2 Graph of the absolute-value function y = f(x) = |x|.

20. Functions and its Graphs Example 3. Constant functions. A function whose range consists of a single number is called a constant function. An example is shown in Figure-3, where f(x) = 3 for every real x. The graph is a horizontal line cutting the Y-axis at the point (0, 3). Figure- 3 Graph of the constant function f(x) = 3.

21. Functions and its Graphs Example 4. Linear functions and affine linear function. Let the function g be defined for all real x by a formula of the form g(x) = ax + b, where a and b are real numbers, then g is called a linear function if b = 0. Otherwise, g is called a affinelinear function. The example, f(x) = x, shown in Figure-1 is a linear function. And, g(x) = 2x - 1, shown in Figure-4 is a affinelinear function. Figure- 4 Graph of the affinelinear function g(x) = 2x - 1.

22. Example 5. Functions and its Graphs The greatest integer function The greatest integer function is defined by f(x) = [x] = The greatest integer less than or equal to x. Figure- 5 shows the graph of f(x) = [x]. Figure- 5 Graph of the greatest integer function is defined by f(x) = [x].

23. Functions and its Graphs Example 6. Polynomial functions. A polynomial function P is one defined for all real a by an equation of the form P(x) = c0 + c1x + c2x2 + c3x3 + ……………+ cnxn The numbers c0 , c1, c2, c3,……………,cnare called the coefficients of the polynomial, and the nonnegative integer n is called its degree (if cn ≠ 0). They include the constant functions and the power functions as special cases. Polynomials of degree 2, 3, and 4 are called quadratic, cubic, and quartic polynomials, respectively. Figure-6 shows a portion of the graph of a quartic polynomial P given by P(x) = ½ x4 – 2x2. Figure- 6 Graph of a quartic polynomial function p(x) = ½ x4 – 2x2.

24. Functions and its Graphs f g are functions k = constant h1= f +g h2= f .g h3= f /g h4= k + f h5= k . f h6= f∘g h7= g∘ f

25. We can define Functions and its Graphs h1= f +g, h2= f .g , h3= f /g, h4= k + f and h5= k . f by h1(x)= (f +g)(x) = f(x) + g(x), h4(x)= (k + f)(x) = k + f(x) , h2(x)= (f .g)(x) = f(x) . g(x), h5(x)= (k . f)(x) = k . f(x) , h3(x)= (f /g)(x) = f(x) / g(x), if f and g are real valued and defined on the same domain X and k is a real number. Also we can define h6= f∘g and h7= g∘ f by h6(x)= (f∘g)(x) = f(g(x)) if co-domain of g = domain of f and h7(x)= (g∘f)(x) = g (f(x)) if co-domain of f = domain of g Generally, f∘g ≠ g∘ f

26. Composition Function If f : X → Y and g : Y → Z then we define a function (g∘ f) : X → Z by (g∘ f)(x) ≡ g(f(x)).

27. Composition Function Example Let f : X → Y and g : Y → Z be define by the following diagrams Then we can compute (g∘ f) : X → Z by its definition: (g∘ f)(a) ≡ g(f(a)) = g(y) = t (g∘ f)(b) ≡ g(f(b)) = g(z) = r (g∘ f)(c) ≡ g(f(c)) = g(y) = t Remark: Let f : X → Y. Then 1Y∘ f = f and f∘ 1X = f where 1X and 1Y are identity functions on X and Y respectively. That is the product of any function and the identity function is the function itself.

28. Functions and its Graphs Theorem: Let f : X → Y, g : Y → Zand h : Z → W.Then (h∘ g)∘ f = h∘ (g∘ f). Theorem Two functions f and g are equal if and only if (a) f and g have the same domain, and (b) f(x) = g(x) for every x in the domain of f.

29. Outline • Functions and its graphs. • One-one, Onto and inverse functions. • Transcendental functions. • Bounded and monotonic functions. • Limits of functions. • Right and left hand limits. • Special limits. • Continuity. • Right and left hand continuity. • Sectional continuity. • Uniform continuity, Semi-continuity, Lipschitz continuity.

30. One-one Function X Y Different elements of X Different elements of Y f is one-one f : X → Y is said to be one-one if f(x) = f(y) implies x = y or, equivalently, x ≠ y implies f(x) ≠ f(y).

31. One-oneFunction Examples (vi) Let X = {1, 2, 3, 4} and Y = {a, b, c, d, e} and f and g be two functions from X into Y given by the following diagrams Here g is a one-one function since different elements of X have different images. Here f is not a one-onefunction since a is the image of two different elements 1 and 2 of X.

32. Functions and its Graphs The Horizontal Line Test This is not the graph of a one- one function. The horizontal line we have drawn cuts the graph twice. This is the graph of a one- one function. All possible horizontal lines will cut this graph only once.

33. Examples One-oneFunction The function f : R → R defined by f(x) = x2is not a one-onefunction since f(2) = f(-2) = 4. Figure- 7 Graph of the function f(x) = x2.

34. Examples One-oneFunction The function f : R → R defined by f(x) = ex is a one-onefunction. Proof: Let f(x) = f(y) then ex = ey i.e. x = y Hence by definition f is a one-onefunction. Figure- 8 Graph of the function f(x) = ex.

35. Examples One-oneFunction The absolute-value function f : R → R defined by f(x) =|x|is not a one-onefunction since f(2) = f(-2) = 2. Figure- 9 Graph of the absolute-value function y = f(x) = |x|.

36. Examples One-oneFunction The identity function f : R → R defined by f(x) = x is a one-onefunction. Proof: Let f(x) = f(y) then x = y Hence by definition f is a one-onefunction. Figure- 10 Graph of the identity function f(x) = x.

37. Onto Function X Y Every element of Y is the image of some element of X f is onto f :X → Y is said to be onto if for every y ϵY, ∃ an element x ϵ X such that y = f(x), i.e. f(X) = Y.

38. Onto Function Examples (vi) Let X = A = {1, 2, 3, 4}, Y = {a, b, c} and B= {a, b, c, d, e} and f and g be two functions from X into Y and A into B respectively given by the following diagrams Here fis a ontofunction since every element of Y appears in the range of f Here g is not a onto function since e ∊ B is not an image of any element of A.

39. Onto Function The Horizontal Line Test This is not the graph of a onto function. the horizontal line drawn above does not cut the graph. This is the graph of a onto function. All possible horizontal lines will cut this graph at least once.

40. Onto Function Examples The function f : R → R defined by f(x) = ex is a not an ontofunction. Proof: Since -2 is an element of the co-domainR then there does not exist any element x in the domainR such that -2 = ex = f(x). Hence by definition f is not a ontofunction. Figure- 11 Graph of the function f(x) = ex.

41. Examples Onto Function The identity function f : R → R defined by f(x) = x is a ontofunction. Proof: Since for every y in the co-domainR, ∃ an element xin the domainR such that y = f(x). Hence by definition f is a ontofunction. Figure- 12 Graph of the identity function f(x) = x.

42. Examples Onto Function The function f : R → R defined by f(x) = x2is not an ontofunction. Proof: Since -2 is an element of the co-domainR then there does not exist any element x in the domainR such that -2 = x2 = f(x). Hence by definition f is not a ontofunction. Figure- 13 Graph of the function f(x) = x2.

43. Inverse Function f is one-one and onto X Y f x y = f(x) f -1

44. Example Let f : X → Y be define by the following diagram Inverse Function X Y Here f is one-one and onto. Therefore f -1, the inverse function, exists. We describe f -1: Y → X by the diagram Y X

45. Inverse Function Theorem on the inverse function. Theorem 1 Let the function f : X → Y be one-one and onto; i.e. the inverse function f -1: Y → X exists. Then the product function (f -1 ∘ f ) : X → X is the identity function on X, and the product function (f ∘ f -1) : Y → Y is the identity function on Y, i.e. (f -1 ∘ f ) = 1X and (f ∘ f -1) = 1Y. Theorem 2 Let f : X → Y and g : Y → X. Then g is the inverse of f, i.e. g = f -1 , if the product function (g∘ f ) : X → X is the identity function on X, and the product function (f ∘ g) : Y → Y is the identity function on Y, i.e. (g∘ f ) = 1X and (f ∘ g ) = 1Y.

46. Outline • Functions and its graphs. • One-one, Onto and inverse functions. • Transcendental functions. • Bounded and monotonic functions. • Limits of functions. • Right and left hand limits. • Special limits. • Continuity. • Right and left hand continuity. • Sectional continuity. • Uniform continuity, Lipschitz continuity.

47. Algebraic and Transcendental Functions Algebraic functions Algebraic functions are functions y = f (x) satisfying an equation of the form p0(x)yn + p1(x)yn-1 + . . . + pn-1(x)y + pn(x) = 0 …………………………… (1) where p0(x) , . . . , pn(x) are polynomials in x. If the function can be expressed as the quotient of two polynomials, i.e., P(x)/Q(x) where P(x) and Q(x) are polynomials, it is called a rational algebraic function; otherwise, it is an irrational algebraic function. Example is an algebraic function since it satisfies the equation (x2 – 2x + 1)y2 + (2x2 – 2x)y + (x2 – x) = 0. Transcendental functions Transcendental functions are functions which are not algebraic; i.e.,they do not satisfy equations ofthe form of Equation (1).

48. The following are sometimes called elementary transcendental functions. Algebraic and Transcendental Functions 1. Exponential function: f (x) = ax, a ≠ 0, 1. 2. Logarithmic function: f (x) = logax, a ≠ 0, 1. This and the exponential function are inverse functions. If a = e = 2.71828 . . . , called the natural base of logarithms, we write f (x) = logex = In x, called the natural logarithm of x. 3. Trigonometric functions: The variable x is generally expressed in radians (π radians = 180∘). For real values of x, sin x and cos x lie between –1 and 1 inclusive. 4. Inverse trigonometric functions. The following is a list of the inverse trigonometric functions and their principal values:

49. Outline • Functions and its graphs. • One-one, Onto and inverse functions. • Transcendental functions. • Bounded and monotonic functions. • Limits of functions. • Right and left hand limits. • Special limits. • Continuity. • Right and left hand continuity. • Sectional continuity. • Uniform continuity, Lipschitz continuity.

50. BoundedFunction Bounded function A function f defined on some set X with real values is called bounded, if the set of its values is bounded. In other words, there exists a real number M < ∞ such that |f(x)| ≤ M or –M ≤ f(x) ≤ M for all x in X. Geometrically, the graph of such a function lies between the graphs of two constant step functions s and t having the values — M and +M, respectively. Figure- 14 Graph of a bounded function. Intuitively, the graph of a bounded function stays within a horizontal band.