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F(x) = | x | = x , for x ≥ 0 = - x , for x ≤ 0

f( Log x) = | Log x | = Log x , for 1 < x ≤ ∞ = 0 for x = 1 = - Log x for 0 < x < 1 = undefined for x ≤ 0

The function f (Log x) exists in (0, ∞). It is continuous at all points. Let us find the derivative from left side and right side of x = 1.

Right side Differential coefficient for x > 1:

Left side differential coefficient for 0< x < 1 :

At x = 1, the derivative from right side is 1 and from left side is -1. So there is no derivative defined for x = 1. Otherwise, it is defined as:

Differential coefficient of f (Log x) : 1/x for x > 1 undefined for x = 1 -1/x for 0 < x < 1 undefined for x <= 0