Question

Prove that the function f given by is notdifferentiable at x = 1.

Answer 1

The function is defined as,

f( x )=| x−1 |

Rewrite the function into two parts:

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

Consider the expression for differentiation of a function by the first principle.

f ′ ( x )= lim h→0 f( x+h )−f( x ) h

For the function to be differentiable, the left hand limit and the right hand limit at point x=1 should be equal.

The left hand limit at x=1 is,

lim h→ 0 − f( 1+h )−f( 1 ) h = lim h→0−p ( 1+h )−( 0 ) h = lim p→0 ( 1+0−p )−( 0 ) 0−p = lim p→0 −p( 1 p −1 ) −p =−1

The right hand limit at x=1 is,

lim h→ 0 − f( 1+h )−f( 1 ) h = lim h→0+p ( 1+h )−( 0 ) h = lim p→0 ( 1+0+p )−( 0 ) 0+p = lim p→0 p( 1 p +1 ) p =1

The right hand limit and the left hand limit are unequal.

Hence, the function f is not differentiable at x=1.