Question

Prove that the greatest integer function defined by is not differentiable at x = 1 and x = 2.

Answer 1

The function is defined as,

f( x )=[ x ]   ∀x∈[ 0,3 )

Rewrite the function into two parts:

f( x )={ 0   ∀x∈[ 0,1 ) 1   ∀x∈[ 1,2 ) 2   ∀x∈[ 2,3 ) }

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 − 1−1 h =0

The right hand limit at x=1 is,

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

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

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

For the function to be differentiable, the left hand limit and the right hand limit at point x=2 should be equal.

The left hand limit at x=2 is,

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

The right hand limit at x=2 is,

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

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

Hence, the function f is not differentiable at x=2.