Tiêu đề 05. CONTINUITY AND DIFFERENTIABILITY.pdf ✅

DEFINITION OF CONTINUITY (i) The continuity of a real function (f) on a subset of the real numbers is defined when the function exists at point c and is given as- lim x → c f(x) = f(c) (ii) A real function (f) is said to be continuous if it is continuous at every point in the domain of f. Consider a function f(x), and the function is said to be continuous at every point in [a, b] including the endpoints a and b. Continuity of “f” at a means, lim x → a f(x) = f(a) Continuity of “f” at b means, lim x → b f(x) = f(b) REMARK: A function f(x) fails to be continuous at x = a for nay of the following reasons. (i) limx→a f(x) exists but it is not equal to f(a) (ii) limx→a f(x) does not exist. (iii) f is not defined at x = a i.e., f(a) does not exist. CONTINUITY AND DERIVABILITY A function f(x) is said to be continuous at x = c, ifLimx→c f(x) = f(c) i.e. f is continuous at x =c if Lim h→0+ f(c − h) = Lim h→0+ f(c+ h) = f(c). If a function f(x) is continuous at x = c, the graph of f(x) at the corresponding point (c, f(c)) will not be broken. But if f(x) is discontinuous at x = c, the graph will be broken when x = c (i) (ii) (iii) (iv) Consider the function ൝ 5 − 2x ; x < 1 3 ; x = 1 x + 2 ; x > 1 Check whether the function is continuous for all x. Solution: Let us check the conditions for continuity. Given f (1) = 3 LHL: limx→1− f(x) = 5 − 2 = 3 RHL: lim x→1+ f(x) = 1 + 2 = 3 LHL = RHL. Also f (1) exists. So the function is continuous at x = 1 Example Discuss the continuity of the function f given by f(x) = x 3 + x 2 − 1 Solution: Clearly f is defined at every real number c and its value at c is c 3 + c 2 − 1 Thus limx→c f(x) = limx→c (x 3 + x 2 − 1) = c 3 + c 2 − 1 Thus limx→c f(x) = f(c) , and hence f is continuous at every real number. This means f is a continuous function. Example CONTINUITY AND DIFFERENTIABILITY CHAPTER – 5
(i), (ii) and (iii) are discontinuous at x = c) (iv) is continuous at x = c) A function f can be discontinuous due to any of the following three reasons: (i) Limx→c f(x) does not exist i.e. Limx→c− f(x) Lim x→c+ f (x) [figure (i)] (ii) f(x) is not defined at x = c [figure (ii)] (iii)Limx→c f(x)  f (c) [figure (iii)] Geometrically, the graph of the function will exhibit a break at x= c. ALGEBRA OF CONTINUOUS FUNCTIONS THEOREM 1: Let f and g be two real functions , continuous at x = a .Let α be real number .Then, (i) f + g is continuous at x = a (ii) f – g is continuous at x = a (iii) αf is continuous at x = a (iv) f.g is continuous at x = a (v) 1 f is continuous at x = a , provided that f(a) ≠ 0 (vi) f g is continuous at x = a , provided that g (a) ≠ 0 THEOREM 2: Let f and g be real functions such that fog is defined .If g is continuous at x =a and f is continuous at g(a) then fog is continuous at x= a CONTINUITY ON AN INTERVAL CONTINUITY ON AN OPEN INTERVAL : A function f(x) is said to be continuous on an open interval (a ,b) iff it is continuous at every point on the interval (a ,b). CONTINUITY ON AN CLOSED INTERVAL : A function f(x) is said to be continuous on a closed interval [a ,b] iff (i) f is continuous on (a , b) (ii) lim x→a+ f(x) = f(a) and , (iii) limx→b− f(x) = f(b) CONTINUOUS FUNCTION: A function f(x) is said to be continuous , if it is continuous at each point of its domain. EVERTWHERE CONTINUOUS FUNCTION: A function f(x) is said to be everywhere continuous if it is continuous on the entire real line ( -∞ , ∞). THEOREM 1: If f and g are two continuous functions on their common domain D , then (i) f + g is continuous on D (ii) f – g is continuous on D (iii) f.g is continuous on D (iv) αf is continuous on D, where α is any real number. (v) f g is continuous on D- { x : g(x) ≠ 0} (vi) 1 f is continuous on D – {x : f(x) ≠ 0 } THEOREM 2: The composition of two continuous functions is a continuous function. THEOREM 3: If f is continuous on its domain D , then |f| is also continuous on D. REMARK: The converse of the above theorem may not be true . THEOREM 4: A constant function is everywhere continuous. THEOREM 5: The identity function is everywhere continuous. Prove that f(x) = tan x is a continuous function. Solution: Given, f(x) = tan x Since, we know that, by trigonometry formulas, tan x = sin x/cos x Thus, f(x) = sin x/cos x Now, this function is defined for all real numbers and cos x ≠ 0, x ≠ (2n+1)π 2 . Sin x and cos x both are continuous functions, therefore, tan x is also continuous, since it is the quotient of sin x/cos x. Example f(x) = ൜ 1 , if x ∈ Z −1 ,if x ∈ R − Z Let a be an arbitrary integer , Then limx→a− f(x) = lim h→0 f(a − h) = lim h→0 −1 = −1 lim x→a+ f(x) = lim h→0 f(a + h) = lim h→0 −1 = −1and f(1) =1 ∴ limx→a− f(x) = lim x→a+ f(x) ≠ f(a) So, f is discontinuous at x = a Now , |f|(x)=|f(x)| = 1 for all x ∈ R . So, |f| is a constant function and hence , it is everywhere continuous. Example
THEOREM 6: A polynomial function is everywhere continuous. COROLLARY: Every rational function is continuous at every point in its domain. THEOREM 7: The modulus function is everywhere continuous. THEOREM 8: The exponential function a x , a> 0 is everywhere continuous. COROLLARY: e x is everywhere continuous. THEOREM 9: The logarithmic function is continuous in its domain. THEOREM 10: The sine function is everywhere continuous. THEOREM 11: The cosine function is everywhere continuous. THEOREM 12: The tangent function is continuous in its domain. THEOREM 13: The cosecant , secant , cotangent are continuous in its domain. REMARK: All inverse trigonometric functions are continuous I their respective domain. DIFFRENTIABILITY AT A POINT y = f(x) y x a – h a a + h P R Q (i) The right hand derivative (R.H.D) of f(x) at x = a denoted by f (a+) is defined by slop of PQ  R.H.D. = f (a+) = h 0 Limit → + f(a+h)−f(a) h , provided the limit exists. (ii) The left hand derivative (L.H.D) of f(x) at x = a denoted by f(a–) is defined by slop of PR  L.H.D. = f  (a– ) = h 0 Limit → + f(a−h)−f(a) −h , provided the limit exists. A function f(x) is said to be differentiable at x = a if f′(a+) = f  (a–) = finite By definition f (a) = h 0 Limit → + f(a+h)−f(a) h CONCEPT OF TANGENT AND ITS ASSOCIATION WITH DERIVABILITY Tangent :- The tangent is defined as the limiting case of a chord or a secant. Slope of the line joining (a,f(a)) and (a + h, f(a + h)) = f(a+h)−f(a) h Prove that f(x) = ඥȁxȁ − x is continuous for all x ≥ 0 Solution: Let g (x) = |x| -x and h(x) = ξx .Clearly , domain(g) = R and domain (h) = [0 , ∞ ). Also , g(x) and h (x) are continuous in their domains. We observe that Domain (hog) = { x∈ Domain (g):g(x) ∈ Domain (h)} ⇒ Domain(hog) = ሼ x ∈ R ∶ ȁxȁ − x ∈ ሾ0 , ∞)} = ሼx ∈ R ∶ x ≥ 0} = ሾ0 , ∞) Example A function f (x) is defined f(x) = ቐ −x 2 ; x ≤ 0 5x − 4 ; 0 < x ≤ 1 4x 2 − 3x ; 1 < x Discuss the differentiability off(x). Solution: The critical points for this function are x = 0, 1 At x =0 L.H.D =lim h→0 −(−h) 2−0 −h = 0 R.H.D = lim h→0 5h−4 h = −∞ Therefore, this function is non-differentiable at x = 0. At x = 1 L.H.D =lim h→0 f(1−h)−f(1) h =lim h→0 5(1−h)−4−1 −h = 5 R.H.D =lim h→0 f(1+h)−f(1) h = lim h→0 4(1+h) 2−3(1+h)−1 h = 5 Since LHL = RHL and LHD = RHD, f (x) is differentiable at x = 1 Example
Slope of tangent at P = f(a) = h 0 Lim → f(a+h)−f(a) h The tangent to the graph of a continuous function f at the point P(a, f(a)) is (i) the line through P with slope f (a) if f (a) exists ; (ii) the line x = a if L.H.D. and R.H.D. both are either  or – . If neither (i) nor (ii) holds then the graph of f does not have a tangent at the point P. In case (i) the equation of tangent is y – f(a) = f(a) (x – a). In case (ii) it is x = a Note: (i) Tangent is also defined as the line joining two infinitesimally close points on a curve. (ii) A function is said to be derivable at x = a if there exist a tangent of finite slope at that point.f(a+) = f(a–) = finite value (iii) y = x3 has x-axis as tangent at origin. (iv) y = |x| does not have tangent at x = 0 as L.H.D RELATION BETWEEN DIFFERENTIABILITY AND CONTINUITY (i) If f (a) exists, then f(x) is continuous at x = a. (ii) If f(x) is differentiable at every point of its domain of definition, then it is continuous in that domain. Note: The converse of the above result is not true i.e. "If 'f' is continuous at x = a, then 'f' is differentiable at x = a is not true. e.g. the functions f(x) = x − 2  is continuous at x = 2 but not differentiable at x = 2. If f(x) is a function such that R.H.D = f(a+) =  and L.H.D. = f(a–) = m. Case -  If  = m = some finite value, then the function f(x) is differentiable as well as continuous. Case -  if m = but both have some finite value, then the function f(x) is non differentiable but it is continuous. Case -  If at least one of the  or m is infinite, then the function is non differentiable but we can not say about continuity of f(x). (i) (ii) (iii) continuous and differentiable |continuous but not differentiable| neither continuous nor differentiable THEOREM : If a function f is differentiable at a point c, then it is also continuous at that point but converse need not to be true.