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f x x Let x be any arbitrary element of R then Right hand limit...

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Calculus

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f x x Let x be any arbitrary element of R then Right hand limit at x x lim f x lim f x h X X Institute of Lifelong Learning University of Delhi lim x h 40 lim x h 2xh 0 0 lim f x x X X Left hand limit at x x lim f x lim f x h lim x h 60 lim x h 2x h x 0 0 lim f x x x x Hence lim f x x 2 Au Since lim f x lim f x x2 X 195 Continuity of Functions Now the value of the function at x x is SITY f x x from equation 1 and 2 we have lim f x f x x X 30 f x f x 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 if x 0 X 0 if x 0 is continuous on R Solution Given function is DELHI xsin if x 0 1 2

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