( x ) = , if , ≠ is not continuous at x = . Solution The function is defined at x = and its value at x = is . When x ≠ , the function is given by a polynomial. Hence, lim ( ) Since the limit of f at x = does not coincide with f ( ), the function is not continuous at x = .
It may be noted that x = is the only point of discontinuity for this function. Example Check the points where the constant function f ( x ) = k is continuous. Solution The function is defined at all real numbers and by definition, its value at any real number equals k . Let c be any real number.
Then c f x = lim c k k Since f ( c ) = k = lim → f ( x ) for any real number c , the function f is continuous at every real number. Example Prove that the identity function on real numbers given by f ( x ) = x is continuous at every real number. Solution The function is clearly defined at every point and f ( c ) = c for every real number c . Also, c f x = lim c x Thus, lim → f ( x ) = c = f ( c ) and hence the function is continuous at every real number.
Having defined continuity of a function at a given point, now we make a natural extension of this definition to discuss continuity of a function. Definition A real function f is said to be continuous if it is continuous at every point in the domain of f . This definition requires a bit of elaboration. Suppose f is a function defined on a closed interval [ a , b ], then for f to be continuous, it needs to be continuous at every point in [ a , b ] including the end points a and b