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


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.