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

Answer

To solve the problem, we need to analyze the functions \( f \) and \( g \) given in the question and determine the points of discontinuity and non-differentiability of the composite function \( f \circ g \).

### Step 1: Analyze the function \( g(x) \)

The function \( g(x) = x - [x] \) represents the fractional part of \( x \). This function is continuous everywhere in the interval \((-1, 1)\) but has discontinuities at integer values. In the interval \((-1, 1)\), the only integer is \( 0 \). Therefore, \( g(x) \) is continuous except at \( x = 0 \).