Question

Find all the points of discontinuity of f defined by .

Answer 1

The function f is defined as ,

f( x )=| x |−| x+1 |

The function f is the difference of two functions f 1 ( x )=| x | and f 2 ( x )=| x+1 |.

The function f 1 ( x ) is defined as,

f 1 ( x )={ x    ∀ x>0 −x  ∀ x≤0 }

And, the function f 2 ( x )is defined as,

f 2 ( x )={ x+1    ∀ x>−1 −x−1  ∀ x≤−1 }

Thus, f is again defined as,

f( x )= f 1 ( x )− f 2 ( x ) f( x )={ −x−( −x−1 )        ∀  x≤−1 −x−( x+1 )          ∀  −1<x≤0 x−( x+1 )            ∀  x>0 } f( x )={ 1                         ∀  x≤−1 −2x−1                ∀  −1<x≤0 −1                       ∀  x>0 }

The left hand limit of the function at x=−1 is,

lim x→− 1 − f( x )= lim x→− 1 − 1 =1 (1)

The right hand limit of the function at x=−1 is,

lim x→− 1 + f( x )= lim x→− 1 + ( −2x−1 ) = lim x→−1+h ( −2x−1 ) = lim h→0 ( −2( −1+h )−1 ) = lim h→0 ( 2−2h−1 )

Solve for the right hand limit.

lim x→− 1 + f( x )=1(2)

The exact value of the function for x=−1is,

f( x=−1 )=1(3)

From equations (1), (2) and (3),

lim x→− 1 − f( x )= lim x→− 1 + f( x )=f( x=−1 )(4)

From equation (4), the condition of continuity of the function f at x=2 is fulfilled.

The left hand limit of the function at x=0 is,

lim x→ 0 − f( x )= lim x→ 0 − ( −2x−1 ) = lim x→0−h ( −2x−1 ) = lim h→0 ( −2( 0−h )−1 ) = lim h→0 ( 2h−1 ) (5)

Solve for the left hand limit.

lim x→ 10 − f( x )=−1(6)

The right hand limit of the function at x=0 is,

lim x→ 0 f( x )= lim x→ 0 + −1 =−1 (7)

The exact value of the function for x=0is,

f( x=0 )=−1(8)

The condition for continuity of the function f at x=0 is fulfilled if left hand limit, right hand limit and the value at the specified points are equal.

From equations (1), (2) and (3),

lim x→ 10 − f( x )= lim x→ 10 + f( x )=f( x=2 )(9)

From equation (4), the condition for continuity of the function f at x=10 is fulfilled.

Hence, function f has no point of discontinuity because it is a continuous function.