. Prove that the function f given by
f(x) = |x – 1|, x€ R
is not differentiable at x = 1.
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Continuity and Differentiability
Differentiability of a Function
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Asked on November 22, 2019 by
Neelam Samariya
Prove that the function f given by f(x)=∣x−1∣,x∈R is not differentiable at x=1.
EASY
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ANSWER
The given function is f(x)=∣x−1∣,x∈R.
It is known that a function f is differentiable at point x=c in its domain if both
lim
h→0
−
h
f(c+h)−f(c)
and lim
h→0
+
h
f(c+h)−f(c)
are finite and equal.
To check the differentiability of the function at x=1,
Consider the left hand limit of f at x=1
lim
h→0
−
h
∣1+h−1∣−∣1−1∣
=lim
h→0
−
h
∣h∣
=lim
h→0
−
h
−h
=−1
Consider the right hand limit of f at x−1
lim
h→0
+
h
∣1+h−1∣−∣1−1∣
=lim
h→0
+
h
h
=1
Since the left and right hand limits of f at x=1 are not equal, f is not differentiable at x=1.