. Hence x = is the only point of discontinuity of f . The graph of the function is given in the Fig . .
Fig . Fig . Example Discuss the continuity of the function defined by f ( x ) = , if , if −+ > Solution Observe that the function is defined at all real numbers except at . Domain of definition of this function is D ∪ D where D = { x ∈ R : x < } and D = { x ∈ R : x > } Case If c ∈ D , then lim ( x + ) = c + = f ( c ) and hence f is continuous in D .
Case If c ∈ D , then lim ( – x + ) = – c + = f ( c ) and hence f is continuous in D . Since f is continuous at all points in the domain of f , we deduce that f is continuous. Graph of this function is given in the Fig . .
Note that to graph this function we need to lift the pen from the plane of the paper, but we need to do that only for those points where the function is not defined. Example Discuss the continuity of the function f given by f ( x ) = , , if ≥ Solution Clearly the function is defined at every real number. Graph of the function is given in Fig . .
By inspection, it seems prudent to partition the domain of definition of f into three disjoint subsets of the real line. Let D = { x ∈ R : x < }, D = { } and D = { x ∈ R : x > } Case At any point in D , we have f ( x ) = x and it is easy to see that it is continuous there (see