hand and left hand limits at x = c coincide, then we say that the common value is the limit of the function at x = c . Hence we may also rephrase the definition of continuity as follows: a function is continuous at x = c if the function is defined at x = c and if the value of the function at x = c equals the limit of the function at x = c. If f is not continuous at c , we say f is discontinuous at c and c is called a point of discontinuity of f . Fig .
Example Check the continuity of the function f given by f ( x ) = x + at x = . Solution First note that the function is defined at the given point x = and its value is . Then find the limit of the function at x = . Clearly lim( ) ( ) Thus ( ) f Hence, f is continuous at x = .
Example Examine whether the function f given by f ( x ) = x is continuous at x = . Solution First note that the function is defined at the given point x = and its value is . Then find the limit of the function at x = . Clearly Thus ( ) f Hence, f is continuous at x = .
Example Discuss the continuity of the function f given by f ( x ) = | x | at x = . Solution By definition f ( x ) = , if , if ≥ Clearly the function is defined at and f ( ) = . Left hand limit of f at is lim (– ) Similarly, the right hand limit of f at is Thus, the left hand limit, right hand limit and the value of the function coincide at x = . Hence, f is continuous at x = .
Example Show that the function f given by f