Question

Let f(x) be defined in the interval [- 2, 2] such that 

f(x) = -1, -2 ≤ x ≤ 0

f(x) = x - 1, 0 < x ≤ 2

and g(x) = f(|x|) + |f(x)|

Test the differentiability of g(x) in (- 2, 2).

Answer 1

Given  that,

RHD (at x = 1) = 2, LHD (at x = 1) = 0

g(x) is not differentiable (at x = 1)

Also, RHD (at x = 0) = 0, LHD (at x = 0) = -1

g(x) is not differentiable (at x = 0)

Hence, g(x) is differentiable for all x ∈ (-2, 2) - {0, 1}