( c ) = [ c ] and hence the function is continuous at all real numbers not equal to integers. Case Let c be an integer. Then we can find a sufficiently small real number r > such that [ c – r ] = c – whereas [ c + r ] = c . This, in terms of limits mean that c − f ( x ) = c – , lim c + f ( x ) = c Since these limits cannot be equal to each other for any c , the function is discontinuous at every integral point.
. . Algebra of continuous functions In the previous class, after having understood the concept of limits, we learnt some algebra of limits. Analogously, now we will study some algebra of continuous functions.
Since continuity of a function at a point is entirely dictated by the limit of the function at that point, it is reasonable to expect results analogous to the case of limits. Theorem Suppose f and g be two real functions continuous at a real number c . Then ( ) f + g is continuous at x = c . ( ) f – g is continuous at x = c .
( ) f . g is continuous at x = c . ( ) f g is continuous at x = c , (provided g ( c ) ≠ ). Proof We are investigating continuity of ( f + g ) at x = c .
Clearly it is defined at x = c . We have lim( )( ) c f g = lim[ ( ) ( )] c f x g x (by definition of f + g ) = lim lim ( ) g x (by the theorem on limits) = f ( c ) + g ( c ) (as f and g are continuous) = ( f + g ) ( c ) (by definition of f + g ) Hence, f + g is continuous at x = c . Proofs for the remaining parts