DIFFERENTIABILITY · Part 8

Example ). Case At any point in D , we have f ( x ) = x and it is easy to see that it is continuous there (see Example ). Fig . Fig . Case Now we analyse the function at x = . The value of the function at is f ( ) = . The left hand limit of f at is – The right hand limit of f at is Thus = f ( ) and hence f is continuous at . This means that f is continuous at every point in its domain and hence, f is a continuous function. Example Show that every polynomial function is continuous. Solution Recall that a function p is a polynomial function if it is defined by p ( x ) = a + a x + ... + a n x n for some natural number n , a n ≠ and a i ∈ R . Clearly this function is defined for every real number. For a fixed real number c , we have c p x p c By definition, p is continuous at c . Since c is any real number, p is continuous at every real number and hence p is a continuous function. Example Find all the points of discontinuity of the greatest integer function defined by f ( x ) = [ x ], where [ x ] denotes the greatest integer less than or equal to x . Solution First observe that f is defined for all real numbers. Graph of the function is given in Fig . . From the graph it looks like that f is discontinuous at every integral point. Below we explore, if this is true. Fig . Case Let c be a real number which is not equal to any integer. It is evident from the graph that for all real numbers close to c the value of the function is equal to [ c ]; i.e., lim [ ] [ ] . Also f