Continuity Advanced Level Pure Mathematics
Advanced Level Pure Mathematics Calculus I 3
Chapter 1 Continuity
3.1 Introduction 2
3.2 Limit of a Function 2
3.3 Properties of Limit of a Function 9
3.4 Two Important Limits 10
3.5 Left and Right Hand Limits 12
3.6 Continuous Functions 13
3.7 Properties of Continuous Functions 16
Prepared by K. F. Ngai Page 1 Continuity Advanced Level Pure Mathematics
3.1 Introduction
1 f (x) x2 f (x) Example is continuous on R. Example x is discontinuous at x = 0. y y 1 y x 2 y x
0 x 0 x
3.2 Limit of a Function
A. Limit of a Function at Infinity
lim f (x) ℓ Defintion Let f(x) be a function defined on R. x means that for any > 0 , there exists X > 0 such that when x > X , f (x) ℓ . y y
l+ l+ l l l
0 X x 0 X x
lim f (x) ℓ N.B.(1) x means that the difference between f(x) and A can be made arbitrarily small when x is sufficiently large. lim f (x) ℓ (2) x means f(x) → A as x → ∞. (3) Infinity, ∞, is a symbol but not a real value.
There are three cases for the limit of a function when x → ∞, 1. f(x) may have a 2. f(x) may approach to 3. f(x) may oscillate or finite limit value. infinity; infinitely and limit value does not exist.
y y y
L 0 x
0 x 0 x
Prepared by K. F. Ngai Page 2 Continuity Advanced Level Pure Mathematics Theorem UNIQUENESS of Limit Value lim f (x) a lim f (x) b a b If x and x , then .
Theorem Rules of Operations on Limits lim f (x) lim g(x) If x and x exist , then lim[ f (x) g(x)] lim f (x) lim g(x) (a) x x x lim f (x)g(x) lim f (x) lim g(x) (b) x x x f (x) lim f (x) lim x lim g(x) 0 x g(x) lim g(x) x (c) x if . lim[kf (x)] k lim f (x) (d) For any constant k, x x . (e) For any positive integer n, lim[ f (x)]n [lim f (x)]n (i) x x
lim n f (x) n lim f (x) (ii) x x
1 lim 0 N.B. x x
2 3x 2x 7 4 2 lim lim( x 1 x ) Example Evaluate (a) x 5x2 x 1 (b) x
Theorem Let f (x) and g(x) be two functions defined on R. Suppose X is a positive real number. f (x) lim g(x) 0 lim f (x)g(x) 0 (a) If is bounded for x > X and x , then x f (x) lim g(x) lim[ f (x) g(x)] (b) If is bounded for x > X and x , then x ; lim f (x) 0 f (x) lim g(x) (c) If x and is non-zero for x > X , and x , then lim f (x)g(x) x .
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sin x x x lim lim e cos x lim(e sin x) Example Evaluate (a) x x (b) x (c) x xex x cos x lim lim (d) x 1 x (e) x x 1
Theorem SANDWICH THEOREM FOR FUNCTIONS Let f(x) , g(x) , h(x) be three functions defined on R. lim f (x) lim h(x) a If x x and there exists a positive real number X such that when x > X , f (x) g(x) h(x) lim g(x) a , then x .
y
y = h(x)
y = f(x) 0 x
(x 1)(x 3) (x 1)(x 3) x 2 Example (a) Show that for x > 0, x 2 . lim[ (x 1)(x 3) x] (b) Hence find x .
1 lim(1 )n e N.B. n n for any positive integer n.
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1 lim(1 )x e Example Evaluate x x by sandwich rule.
1 1 lim(1 ) x e lim(1 y) y e Theorem x x y0 x2 1 x 2 1 1 lim(1 x) x lim 1x x0 2 lim x Example Evaluate (a) (b) x x 1 (c) x1
1 1 x lim (1 ) e lim(1 x) x e N.B. 1. x x 2. x0 . 1 2 x lim(1 3x) x lim(1 ) Exercise (a) x0 (b) x x
cot x x 1 lim(1 tan x) lim( ) x (c) x (d) x x 1 Prepared by K. F. Ngai Page 5 Continuity Advanced Level Pure Mathematics
B. Limit of a Function at a Point
lim f (x) ℓ Definition Let f(x) be a function defined on R. xa means that for any > 0 , there exists > 0 such that when 0 x a , f (x) ℓ . y y = f(x)
f(x) l
0 a x x
lim f (x) ℓ N.B. (1) xa means that the difference between f(x) and A can be made arbitrarily small when x is sufficiently close to a. lim f (x) f (a) (2) If f(x) is a polynomial , then xa . lim f (x) f (a) (3) In general , xa . lim f (x) (4) f(x) may not defined at x = a even through xa exists.
lim(x 2 2x 5) 32 2(3) 5 8 limcos x cos 1 Example x3 , x .
3 if x 5 f (x) lim f (x) 3 Example Let 1 if x = 5 , then x5 but f(5) = 1 , lim f (x) f (5) so x5 .
2x 1, x 2 f (x) . Example Consider the function 4, x 2 f (x) x 2 lim f (x) lim(2x 1) 5 f (2). is discontinuous at since x2 x2
x 2 4 (x 2)(x 2) f (x) lim f (x) lim lim(x 2) 4 Example Let x 2 , then x2 x2 x 2 x2 .
Prepared by K. F. Ngai Page 6 Continuity Advanced Level Pure Mathematics But f(2) is not defined on R. lim f (x) l N.B. xa is equivalent to : lim [ f (x) ℓ] 0 (i) xa ; lim f (x) ℓ 0 (ii) xa ; lim f (x) ℓ lim f (x); (iii) xa xa lim f (a h) ℓ (iv) h0 .
Theorem Rules of Operations on Limits Let f (x) and g(x) be two funcitons defined on an interval containing a , possibly except a . lim f (x) h lim g(x) k If xa and xa , then lim f (x) g(x) h k (a) xa lim f (x)g(x) hk (b) xa f (x) h lim (if k 0) (c) xa g(x) k ,
x3 1 (1 x)n 1 lim lim Example Evaluate (a) x1 x 1 (b) x0 x 2 x lim lim ( x2 x x) (c) x2 3 x2 5 (d) x
1 x 1 x lim lim Example Evaluate (a) x3 2 2x 2 (b) x1 2 2x 2 x 2 1 x lim lim (c) x2 x 2 2 2 (d) x1 1 3 x
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sin x tan x 2 x ( ,0) (0, ) 2 Example (a) Prove that for 2 2 , 1 cos x cos x . sin x tan x lim 2 (b) Hence, deduce that x0 1 cos x .
3.3 Properties of Limit of a Functon
Theorem Uniqueness of Limit Value lim f (x) h lim f (x) k h k If xa and xa , then .
Theorem Heine Theorem
lim f (x) l lim f (xn ) l where xn a as n xa n .
Prepared by K. F. Ngai Page 8 Continuity Advanced Level Pure Mathematics 1 limsin Example Prove that x0 x does not exist.
x3 ax 2 x 4 lim a Example If x1 x 1 exists, find the value of and the limit value.
Assignment Ex.3A (1-6)
3.4 Two Important Limits
sin x lim 1 Theorem x0 x
sin x cos x x tan x lim lim lim lim N.B . (1) x x x x (2) x0 sin x x0 x cos x 1 lim limsin (3) x0 x (4) x0 x
Prepared by K. F. Ngai Page 9 Continuity Advanced Level Pure Mathematics
Example Find the limits of the following functions: tan x 1 cos x lim lim (a) x0 x (b) x0 x 2 sin 4x lim(x )sec x lim x x0 (c) 2 2 (d) sin 5x
Example Find the limits of the following functions: sin(x 1) cosax cosbx 3sinx sin 3x lim lim lim (a) x1 x 2 3x 2 (b) x0 x 2 (c) x0 x3
0 Example Let be a real number such that 2 . sin 2sin cos By using the formula 2k 2k1 2k1 , show that θ θ θ θ sinθ 2n1 cos cos ...cos sin 2 22 2n1 2n-1 . x x x lim(cos cos ...cos ) Hence, find the limit value n 2 22 2n
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Assignment Ex. 3B (1-2)
3.5 Left and Right Hand Limits
lim f (x) l iff lim f (x) lim f (x) l Theorem xa xa xa .
Example Show that each of the following limits does not exist.
x 1 lim x 2 lim lime x (a) x2 (b) x0 x (c) x0
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1 1 lim(a x b x ) x a b 0 Example By Sandwich rule, show that x0 does not exist for . b a b 0 1 a b 0 x 0 Solution If then a If and then b 1 1 b 1 a 1 x 0 ( ) x 1x 1 ( ) x 1 ( ) x 1 If then a a b 1 1 b 1 1 1 a 1 a (a x b x ) x a{1 ( ) x }x a 2 x b (a x b x ) x b{( ) x 1}x b 2 x a b lim 2 x 1 lim b 2 x b As x0 , by sandwich rule, As x0 , by sandwich rule, 1 1 1 1 lim(a x b x ) x a lim(a x b x ) x b we have x0 we have x0
1 1 a b, lim(a x b x ) x Since x0 does not exist. ( Why ? )
Assignment Ex. 3C (1-3)
3.6 Continuous Functions
lim f (x) f (a) N.B. In general, xa .
Definition Let f(x) be a function defined on R. f(x) is said to be continuous at x = a if and only if lim f (x) f (a) xa .
N.B. This is equivalent to the definition : A function f(x) is continuous at x = a , if and only if (a) f(x) is well-defined at x = a , i.e. f(a) exists and f(a) is a finite value, lim f (x) lim f (x) f (a) (b) xa exists and xa . lim f (a h) f (a) Remark Sometimes, the second condition may be written as h0 Prepared by K. F. Ngai Page 12 Continuity Advanced Level Pure Mathematics 1 xsin , x 0 f (x) x Example Show that the function 0 , x = 0 is continuous at x = 0.
Definition A function is discontinuous at x = a iff it is not continuous at that point a. y There are four kinds of discontinuity: y = f(x) f(a) (1) Removable discontinuity: o lim f (x) xa exists but not equal to f(a). x 0 a 2x 1, x 2 f (x) Example Show that 4 , x = 2 is discontinuous at x = 2.
lim f (x) lim(2x 1) 5 f (2) Solution Since f(2) = 4 and x2 x2 , so f(x) is discontinuous at x = 2.
1 cos x , x 0 f (x) x 2 Example Let a , x = 0 . lim f (x) (a) Find x0 . (b) Find a if f(x) is continuous at x = 0.
y (2) Jump discontinuity : y = f(x) lim f (x) lim f (x) xa xa o
0 a x
Example Show that the function f(x) = [x] is discontinuous at 2.
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Example Find the points of discontinuity of the function g(x) = x [x].
x3 2x 1, x 0 f (x) Example Let sin x, x 0 . Show that f(x) is discontinuous at x = 0.
(3) Infinite discontinuity: y
lim f (x) or lim f (x) xa xa y = f(x) i.e. limit does not exist. 0 a x
1 f (x) Example Show that the function x is discontinuous at x = 0.
y
Prepared by K. F. Ngai Page 14 Continuity Advanced Level Pure Mathematics
1 (4) At least one of the one-side limit does not exist. f (x) e x lim f (x) or lim f (x) xa xa do not exist.
1 1 lim e x 0 and lim e x x Example Since x0 x0 ,
1 so the function f (x) e x is discontinuous at x = 0.
Definition (1) A function f(x) is called a continuous function in an open interval (a, b) if it is continuous at every point in (a, b).
(2) A function f(x) is called a continuous function in an closed interval [a, b] if it is continuous at lim f (x) f (a) lim f (x) f (b) every point in (a, b) and xa and xb . (1) (2) y y y
o
o a b x a b x a b x f(x) is continuous f(x) is continuous f(x) is continuous on (a, b). on (a, b). on [a, b].
f (x) lim f (x) f (a) lim f (a h) f (a) N.B. (1) is continuous at x = a xa h0 . f (x) lim f (x h) f (x) (2) is continuous at every x on R h0 for all x R. Example Let f (x) be a real-value function such that (1) f (x y) f (x) f (y) for all real numbers x and y , (2) f (x) is continuous at x 0 and f (0) 1. Show that f (x) is continuous at every x R.
3.7 Properties of Continuous Functions
Prepared by K. F. Ngai Page 15 Continuity Advanced Level Pure Mathematics A. Continuity of Elementary Functions Theorem Rules of Operations on Continuous Functions If f(x) and g(x) are two functions continuous at x = a , then so are f (x) g(x) , f (x)g(x) f (x) and g(x) provided g(a) 0.
1 cos x h(x) f (x) 1 cos x g(x) x 2 Example Let x 2 . Since and are two functions continuous everywhere, but g(0) = 0, so h is continuous everywhere except x = 0.
1 cos x h(x) f (x) 1 cos x g(x) x 2 1 ( 0) Example Let x 2 1 . Since and are two functions continuous everywhere, so h is continuous everywhere.
Theorem Let g(x) be continuous at x = a and f(x) be continuous at x = g(a), then f。g is continuous at x = a.
x sin x Example sin e , e are continuous functions.
2 Example Show that f (x) cos x is continuous at every x R.
x | x | x if x 0 f (x) g(x) 2 Example Let f and g be two functions defined as 2 for all x and x if x 0 . For what values of x is f [g(x)] continuous?
lim f (x) A sin x, cos x, 1 x Theorem Let x and g(x) be an elementary function (such as sin x, e , ln x, ... etc.) for which g(A) is defined, then
lim g[ f (x)] g[lim f (x)] g(A) x x
limcos(tanx) cos(lim tanx) Example x1 x1
Prepared by K. F. Ngai Page 16 Continuity Advanced Level Pure Mathematics ln(1 x) lim Example Evaluate x x .
sinx lim tan ( ) Example Evaluate x0 x .
B. Properties of Continuous Functions
(P1) If f(x) is continuous on [a, b], then f(x) is bounded on [a, b].
(P2) But f(x) is continuous on (a, b) cannot imples that f(x) is bounded on (a, b).
(1) y c < f(x) < d (2) y f(x) is not bounded d on (a, b).
c 0 a b x
0 a b x
Example f(x) = cot x is continuous on (0, ) , but it is not bounded on (0, ).
(P3) If f(x) is continuous on [a, b], then it will attain an absolute maximum and absolute minimum on [a, b].
Prepared by K. F. Ngai Page 17 Continuity Advanced Level Pure Mathematics i.e. If f(x) is continuous on [a, b], there exist x1 [a,b] such that f(x1 ) f(x) and x2 [a,b] such
that f(x2 ) f(x) for all x[a,b] . x1 is called the absolute maximum of the function and x2 is called the absolute minimum of the function. no. abs. max. on [a, b] y o abs. max. on [c, d] | o
0 a b c d abs. min. on [c, d] abs. min. on [a, b]
(P4) If f(x) is continuous on [a, b] and f(a)f(b) < 0 , then there exists c [a, b] such that f(c) = 0.
y y
0 a b x 0 a b x Example Let f (x) cos x 0.6 which is continuous on [0, 2 ] and f(0) = 0.4 and f ( 2 ) 0.6 . Hence, there exists a real number c [0, 2 ] such that f(c) = 0.
Example Prove that the equation x e x 2 has at least one real root in [0, 1].
3 Example Let f (x) x 2x .
Prepared by K. F. Ngai Page 18 Continuity Advanced Level Pure Mathematics (a) Show that f(x) is a strictly increasing function. (b) Hence, show that the equation f (x) 0 has a unique real root.
(P5) Intermediate Value Theorem If f(x) is continuous on [a, b], then for any real number m lying between f(a) and f(b) , there corresponds a number c [a, b] such that f(c) = m. y
f(b) m
f(a) 0 a c b x
1 (P6) Let f(a) = c and f(b) = d, if f is continuous and strictly increasing on [a, b], then f is also continuous and strictly increasing on [c, d] ( or [d, c] ).
2 1 Example f (x) x is strictly increasing and continuous on [0, 5], so f (x) x is also strictly increasing and continuous on [0, 25].
1 1 Example f (x) cos x is strictly decreasing and continuous on [0,π], so f (x) cos x is also strictly decreasing and continuous on [1, 1].
Prepared by K. F. Ngai Page 19 Continuity Advanced Level Pure Mathematics Example (a) Suppose that the function satisfies f(x+y) = f(x) + f(y) for all real x and y and f(x) is continuous at x = 0. Show that f(x) is continuous at all x. 1 sin x 1 sin x if x 0 f (x) x (b) A function 0 if x = 0 . Is f(x) continuous at x = 0? If not , how can we redefine the value of f(x) at x = 0 so that it is continuous at x = 0?
Exercise Example Let f(x) be a continuous function defined for x > 0 and for any x , y > 0, f(xy) = f(x) + f(y). (a) Find f(1).
r (b) Let a be a positive real number. Prove that f (a ) r f (a) for any non-negative rational number.
(c) It is known that for all real numbers x, there exists a sequence {xn} of rational numbers such
lim xn x that x . x (i) Show that f (a ) x f (a) for all x > 0, where a is a positive real constant. (ii) Hence show that f (x) c ln x for all x > 0, where c is a constant.
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Example Let f be a real-valued continuous function defined on the set R such that f(x+y) = f(x) + f(y) for all x , y R. (a) Show that (i) f(0) = 0 (ii) f(x)= f(x) for any x R. (b) Prove that f(nx) = nf(x) for all integers n. Hence show that f(r) = rf(1) for any rational number r.
(c) It is known that for all x R, there exists a sequence {an }of rational numbers such that
lim an x n . Using (b), prove that there exists a constant k such that f(x) = kx for all x R.
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Example Let R denote the set of all real numbers and f : R R a continuous function not identically zero such that f(x+y) = f(x)f(y) for all x , y R. (a) Show that (i) f(x) 0 for any x R. (ii) f(x) > 0 for any x R. (iii) f(0) = 1 1 (iv) f (x) f (x) r (b) Prove that for any rational number r, f (rx) f (x) . x Hence prove that there exists a constant a such that f (x) a for all x R.
Assignment Ex.3D (1-9). Revision Exercise (1-6).
Prepared by K. F. Ngai Page 22