lim f x x X X Left hand limit atx x lim f x lim f x h X lim f x x X X Continuity of Functions lim x h lim x h 2x h 20 x 0 0 RSITY OF DELE Since lim f x lim f x x Hence lim f x x Now the value of the function at x x is f x x from equation 1 and 2 we have lim f x f x x f x 0 Thus f x x is continuous at x x Since is an arbitrary element of R Therefore f x is continuous at each point of R Hence f x is continuous on R Example 4 Show that the function f x defined by Exfar if x 0 if x 0 is continuous on R Solution Given function is 1 xsin if x 0 X 0 if x 0 N