$Let the functions : (- 1, 1) \to ℝ and g : (- 1, 1) \to (- 1, 1) be defined by f (x) = | 2 x - 1 | + | 2 x + 1 | and g (x) = x - [x]$
$where [x] denotes the greatest integer less than or equal to x . Let f \circ g : (- 1, 1) \to ℝ be the composite$
$function defined by (f \circ g) (x) = f (g (x)) . Suppose c is the number of points in the interval (- 1, 1) at$
$which f \circ g is NOT continuous, and suppose d is the number of points in the interval (- 1, 1) at$
$which f \circ g is NOT differentiable. Then the value of c + d is$

Difficult

Solution

$\begin{aligned} f (x) = | 2 x - 1 | + | 2 x + 1 | \\ g (x) = {x} \\ f (g (x)) = | 2 {x} - 1 | + | 2 {x} + 1 | \\ = \{\begin{array}{cc} 2 & {x} \leq 1 / 2 \\ 4 {x} & {x} > 1 / 2 \end{array} \end{aligned}$
$\begin{array}{l} discontinuous at x = 0 \Rightarrow c = 1 \\ Non differential at x = - 1 / 2, 0, 1 / 2 \Rightarrow d = 3 \\ ∴ c + d = 4 \end{array}$

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