Example 12 Discuss the continuity of the function defined by x 2 if x 0 x 2 if x 0 Solution Observe that the function is defined at all real numbers except at 0 Domain of definition of this function is D D where D x R x 0 and D x R x 0 Case 1 If ce D then lim f x lim x 2 X C X C f x c 2 f c and hence fis continuous in D Case 2 If c E D then lim f x lim x 2 X C X C c 2 f c and hence fis continuous in D Since fis continuous at all points in the domain off we deduce that f is continuous Graph of this function is given in the Fig 5 6 Note that to graph this function we need to lift the pen from the plane of the paper but we need to do that only for thos defined Example 13 Discuss the continuity of the function f given by f x x x if x 0 if x 0 Solution Clearly the function is defined at every real number Graph of the function is given in Fig 5 7 By inspection it seems prudent to partition the domain of definition of finto three disjoint subsets of the real line 112 Let D x R x 0 D 0 and D x R x 0 MATHEMATICS X 3 2 1 The right hand limit of fat 0 is 2 4 1 1 Rationalised 2023 24 y Fig 5 6 points where the function is not Fig 5 7 Case 1 At any point in D we have f x x and it is easy to see that it is continuous there see Example 2 2 2 1 1 0 Case 2 At any point in D3 we have f x x and it is easy to see that it is continuous there see Example 6 lim f x lim x 0 0 x 0 x 0 Persiane lim f x lim x 0 x 0 x 0 X X Case 3 Now we analyse the function at x 0 The value of the function at 0 is f 0 0 The left hand limit of fat 0 is Thus lim f x 0 f 0 and hence fis continuous at 0 This means that fis x 0 continuous at every point in its domain and hence fis a continuous function Example 14 Show that every polynomial function is continuous