# Why /x-1/ is continuous but not differentiable at x = 1

2
by ravee108ravee108

2015-03-09T09:38:52+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
As you can see from the graph, It is continuous at x=1 and its value is 0.
But it is Not differentiable at x=1 as the graph is not smooth at x=1. It is changing its slope at that point.
For x<1, slope is -1 and for x>1, slope is +1. So at x=1, slope is not defined. Thus it is not differentiable at x=1.
2015-03-09T15:01:08+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
y = f(x) = | x - 1 |
=  x - 1   , if  x > 1
=  0      , if x = 1
=  1 - x  , if  x < 1

For continuity of the function at x = 1,  the right limit of the function as x -> 1⁺ from right side (values a little higher than 1) must be equal to the left limit of the function, as x -> 1⁻  (from values a little less than 1).  The right limit is :

The left limit is :

The values of left and right limits are equal and hence, the function is continuous.
===========================
the definition of derivative is

The function must be continuous and then the left derivative and right derivative must exist and then be equal.  Then the derivative at the function exists.

Right derivative is :

Left derivative is :

Since the left and right derivatives are not equal the function given, is not derivable at x = 1.