# 2.5 Continuity.pdf - 2.5 Continuity Consider the function f...

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2.5 Continuity Consider the function f with the following graph. 1. lim x →- 5 f ( x ) = f ( - 5) = 1 . 5 . 2. f ( - 1) = 3 , but lim x →- 1 f ( x ) does not exist. 3. lim x 1 f ( x ) = - 2 , but f (1) is undefined. 4. lim x 2 f ( x ) = 2 6 = 3 = f (2) In 1 , lim x →- 5 f ( x ) = f ( - 5) = 3 , but in 2 , 3 and 4 , the limit and the value are not equal. We say the function f is continuous at x = - 5 and discontinuous at x = - 1 , 1 , 2 . Continuity of a Function at a Number Definition 1 ( Continuity of a Function at a Number ) . A function f is continuous at a num- ber x 0 if lim x x 0 f ( x ) = f ( x 0 ) Otherwise, we say f is discontinuous at a or f has a discontinuity at x 0 . Note: f is continuous at x 0 , if and only if the following three properties. 1. f ( x 0 ) exists. i.e. x 0 is in the domain of f . 2. lim x x 0 f ( x ) exists. 3. The limit equals the value. i.e. lim x x 0 f ( x ) = f ( x 0 ) . I If any one of these fails, then f is discontinuous at x 0 . I f is continuous at x = x 0 means that the graph of f does not have a hole or a jump or a vertical asymptote when x = x 0 . 1
Continuity: Examples Example 1 . For the function f whose graph if given below, find all points of discontinuity between x = - 10 and x = 10 . Example 2 . Find all the points of discontinuity of f ( x ) = x , if there are any.
x 2 - 4
Continuity: More Examples Example