Limits, Continuity and Differentiability

Concept:The function is a sum of a continuous absolute-value term and a discontinuous greatest-integer term.Explanation:Let $f(x) = g(x) + h(x)$, where $g(x) = (|x| + |x-1|)\sin x$ and $h(x) = [x \sin x]$.$g(x)$ is continuous in $(-\pi/2, \pi/2)$ because absolute values and sine are continuous.$h(x)$ is discontinuous whenever $x\sin x$ is an integer, due to the jump of the greatest-integer function.Find where $x\sin x = 0$: gives $x=0$. At $x=0$, $h(x)=0$, and the function does not jump, so $f$ is continuous at $0$.Find where $x\sin x = 1$: there is a point $x_1$ in $(0,\pi/2)$ and its symmetric $-x_1$.At these two points, $[x\sin x]$ jumps from $0$ to $1$, so $f$ is not continuous there. Hence $\alpha = 2$.Differentiability fails at the discontinuity points $x_1$ and $-x_1$ because the function is not continuous there.Also check points where $g(x)$ might be non-differentiable: $x=0$ and $x=1$ due to absolute values.At $x=0$, left and right derivatives match, so differentiable.At $x=1$, left derivative is $\cos 1$, right derivative is $\cos 1 + 2\sin 1$, which are not equal, so $f$ is not differentiable at $x=1$.Thus points of non-differentiability are $x_1$, $-x_1$, and $1$, giving $\beta = 3$.Answer:$\alpha + \beta = 2 + 3 = 5$.