Again, we wish to emphasise that – ∞ is NOT a real number and hence the left hand limit of f at does not exist (as a real number). The graph of the reciprocal function given in Fig . is a geometric representation of the above mentioned facts. Fig .
Example Discuss the continuity of the function f defined by f ( x ) = , if , if − > Solution The function f is defined at all points of the real line. Case If c < , then f ( c ) = c + . Therefore, lim lim( ) Thus, f is continuous at all real numbers less than . Case If c > , then f ( c ) = c – .
Therefore, ( x – ) = c – = f ( c ) Thus, f is continuous at all points x > . Case If c = , then the left hand limit of f at x = is – – lim ( ) = + The right hand limit of f at x = is lim ( ) = − = − Since the left and right hand limits of f at x = do not coincide, f is not continuous at x = . Hence x = is the only point of discontinuity of f . The graph of the function is given in Fig .
. Example Find all the points of discontinuity of the function f defined by f ( x ) = , if , , if − > Solution As in the previous example we find that f is continuous at all real numbers x ≠ . The left hand limit of f at x = is – lim ( ) = + The right hand limit of f at x = is lim ( ) = − = − Since, the left and right hand limits of f at x = do not coincide, f is not continuous at x =