. Continuity of f at a means a f x = f ( a ) and continuity of f at b means – b f x = f ( b ) Observe that lim a f x and lim b f x do not make sense. As a consequence of this definition, if f is defined only at one point, it is continuous there, i.e., if the domain of f is a singleton, f is a continuous function. Example Is the function defined by f ( x ) = | x |, a continuous function?
Solution We may rewrite f as f ( x ) = , if , if ≥ By Example , we know that f is continuous at x = . Let c be a real number such that c < . Then f ( c ) = – c . Also c f x = lim ( – (Why?) Since lim c f x , f is continuous at all negative real numbers.
Now, let c be a real number such that c > . Then f ( c ) = c . Also c f x = lim c x = (Why?) Since lim c f x , f is continuous at all positive real numbers. Hence, f is continuous at all points.
Example Discuss the continuity of the function f given by f ( x ) = x + x – . Solution Clearly f is defined at every real number c and its value at c is c + c – . We also know that c f x lim ( ) c x Thus lim c f x , and hence f is continuous at every real number. This means f is a continuous function.
Example Discuss the continuity of the function f defined by f ( x ) = x , x ≠ . Solution Fix any non zero real number c , we have Also, since for c ≠ , , we have lim c f x and hence, f