In the definition of limits we used the inequality 0 x x whereas in the definition of continuity we have used the inequality x x

Question

In the definition of limits we used the inequality 0 x x whereas in the definition of continuity we have used the inequality x x The reason is that in case of limits the function may or may not be defined at x but for the continuity of a function at the function must be defined on to Institute of Lifelong Learning University of Delhi lim f x f x X I Q 1 3 1 Continuity from the Right at a Point A function f X R is said to be right continuous or continuous form the right at a point ex if the right hand limit of f exists at x and equal to f x i e lim f x f x X X Continuity of Functions 3 2 Continuity from the Left at a Point A function f X R is said to be Left continuous or continuous form the Left at a point x X if the Left hand limit of f exists at x and equal to f x i e pg 4 4 Continuous Function A function f X R is said to be continuous on X if and only if f is continuous at each point of X Example 1 Show that the function f x is not continuous at x 0 Solution Given function is Moreover we know that f x since f x is not defined for x 0 therefore it cannot be continuous at x 0 Exfa 1 lim f x lim x 0 0X does not exist in R Hence f x is not continuous at x 0 Value Addition Note If the limit of a function f XR does not exist at a point x EX in R then the function f X R cannot be continuous at x Institute of Lifelong Learning University of Delhi pg 5

Kunduz · Accepted Answer

Click to see the answer