Let f(x) =
-1, -2 ≤ x < 0
2x, 0 ≤ x ≤ 2
and g(x) = |f(x)| + f(|x|). Then, in the interval (-2, 2), g is:
differentiable at all points
not differentiable at two points
not continuous
not differentiable at one point

|f(x)| = {
1, -2 ≤ x < 0
1-x^2, 0 ≤ x < 1
x^2, 1 ≤ x ≤ 2
}
f(|x|) = {
-1, -2 ≤ x < 0
2x, 0 ≤ x ≤ 2
}
g(x) = |f(x)| + f(|x|) = {
0, -2 ≤ x < 0
1 - x^2 + 2x, 0 ≤ x < 1
x^2 + 2x, 1 ≤ x ≤ 2
}
In the interval (-2, 2), g(x) is not differentiable at x = 0 and x = 1.
Therefore, g(x) is not differentiable at two points.