is a continuous function. Solution Define g by g ( x ) = – x + | x | and h by h ( x ) = | x | for all real x . Then ( h o g ) ( x ) = h ( g ( x )) = h ( – x + | x |) = | – x + | x || = f ( x ) In Example , we have seen that h is a continuous function. Hence g being a sum of a polynomial function and the modulus function is continuous.
But then f being a composite of two continuous functions is continuous. EXERCISE . . Prove that the function f ( x ) = x – is continuous at x = , at x = – and at x = .
. Examine the continuity of the function f ( x ) = x – at x = . . Examine the following functions for continuity.
(a) f ( x ) = x – (b) f ( x ) = x − , x ≠ (c) f ( x ) = , x ≠ – (d) f ( x ) = | x – | . Prove that the function f ( x ) = x n is continuous at x = n , where n is a positive integer. . Is the function f defined by , if , if > = continuous at x = ?
At x = ? At x = ? Find all points of discontinuity of f , where f is defined by . , if , if > = .
| | , if , if < , if ≤− ≥ . | |, if , ≠ = . , if | | , = − ≥ . , if , if ≥ = .