Hence f x is continuous on R Institute of Lifelong Learning University of Delhi I Q 2 I Q 3 Theorem 1 Statement Only A function f X R is said to be continuous at a point X EX if and only if for given any E neighborhood V f x of f x there exist a 8 neighborhood V x of xo such that if x is any point of XV x then f x belongs to V f x is f xV x V f x Continuity of Functions 5 Discontinuity of a Function at a Point Let X be a non empty subset of R let f X R and let X Then the function f x is said to be discontinuous at if and only if it is not continuous at 5 1 Types of Discontinuity Let f be a function defined as f X R and let x eX and let left hand limit right hand limit limit of the function and the value of the function at xare denoted by lim f x lim f x lim f x and f x 195 respectively Then I Removable Discontinuity at x let at s The function f X R is said to have the removable discontinuity at X EX if the limit of the function exist at x but not equal to the value of the function at x i e lim f x f x X pg 8 efa II Discontinuity of the First Kind at x The function f X R is said to have a discontinuity of first kind at x if both the left hand limit and right hand limit exist at x but are not equal to each other i e lim f x lim f x 294 Institute of Lifelong Learning University of Delhi Continuity of Functions III Discontinuity of the First Kind from the left pg 9 9