88AE
Solution 88AE Step-1 A continuous function can be formally defined as a f unction f : x y ,where the preimage of every open set in y is open in x. More concretely, a function f(x) in a single variable x is said to be continuous at point x if, 0 1. If f(x 0 is defined, so that x 0 is in the domain of ‘ f’. 2. lim f(x) exists for x in the domain of f. x x0 3. lim f(x) = f( x ). x x0 0 Left continuous : lim f(x) = f(a) , then f(x) is called a left continuous at x=a. xa Right continuous : lim xa)+= f(a) , then f(x) is called a right continuous at x=a. If , lim f(x) = f(a) = lim f(x) , then f(x) is called a continuous function at x=a. xa xa+ If , f(x) is not continuous at x =a means , it is discontinuous at x=a. Step-2 For any real number x the absolute value or modulus of x is denoted by x . | | | |= { x , if x 0 = { -x , if x 0. The graph of x | |; f(x) = | | It makes a right angle at (0,0) It is an even function. Its domain is the real numbers R, and its range is the non -negative real numbers : [ 0 , +). Step-3 The given function is f(x) = |x2| , at x=2. x2 |x2| Left hand limit : lim f(x) lim x2 x2 x2 (x2) = (x2) , since x | 2 = |(x-2) ,if x 2. = -1. Therefore, lim f(x) = -1. x2 Right hand limit : lim f(x) = lim |2| x2+ x2+ x2 = (x2), since x 2 = (x-2) ,if x 2 (x2) | | = 1. Therefore, lim f(x) = 1. + x2 Hence , lim f(x) = -1 = / 1 = lim f(x) x2 x2 + Therefore , f(x) is not continuous at x =2. Here , lim f(x) and lim f(x) both exists and are not equal to then we say that f has a jump + x2 x2 discontinuity at x=2. NOTE ; Jump discontinuity; It is caused when the function jumps from one point to another in its graph.
