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.