Let $f: \left[-\frac{1}{2}, 2\right] \to \mathbb{R}$ and $g: \left[-\frac{1}{2}, 2\right] \to \mathbb{R}$ be functions defined by $f(x) = [x^{2}-3]$ and $g(x) = |x| f(x) + |4x-7| f(x)$, where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in \mathbb{R}$. Then

$\text{Given: } f(x) = \lfloor x^2 - 3 \rfloor,\quad g(x) = \big(|x| + |4x-7|\big) f(x)$ $\text{domain } \left[-\dfrac{1}{2}, 2\right];\quad [y] = \lfloor y \rfloor$ $\text{To find: which of (1)--(4) are TRUE}$ $1.\ \text{discontinuities of } f \text{ (floor jumps)}$ $2.\ g = h\,f \text{ with } h = |x| + |4x-7|$ $3.\ \text{non-differentiability points of } g$ $f(x) = \lfloor x^2 - 3 \rfloor \text{ jumps where } x^2 - 3 \text{ crosses an integer}$ $x \in \left[-\dfrac{1}{2}, 2\right] \Rightarrow x^2 \in [0,4] \Rightarrow x^2 - 3 \in [-3, 1]$ $x^2 = 1, 2, 3, 4 \Rightarrow x = 1, \sqrt{2}, \sqrt{3}, 2$ $\text{at } x = 0:\ x^2 - 3 \text{ has a minimum, no integer crossing}$ $f \text{ discontinuous at } x = 1, \sqrt2, \sqrt3, 2 \Rightarrow \text{(2) TRUE},\ \text{(1) FALSE}$ $g(x) = h(x)\,f(x), \qquad h(x) = |x| + |4x-7|$ $h(x) = |x| + |4x-7| > 0 \text{ for all } x$ $\text{kinks of } h:\quad x = 0 \quad \text{and} \quad x = \dfrac{7}{4} = 1.75$ $\text{at } x = 1, \sqrt2, \sqrt3:\ h \neq 0 \Rightarrow g \text{ not differentiable } (3 \text{ points})$ $x = \tfrac{7}{4}:\ \sqrt3 < \tfrac74 < 2 \Rightarrow f(x) = \lfloor x^2-3 \rfloor = 0$ $\Rightarrow g \equiv 0 \text{ near } \tfrac74 \Rightarrow g \text{ IS differentiable there}$ $x = 0:\ f(0) = -3 \neq 0 \text{ with } |x| \text{ kink} \Rightarrow g \text{ not differentiable}$ $g \text{ not differentiable at } 0, 1, \sqrt2, \sqrt3 \Rightarrow 4 \text{ points} \Rightarrow \text{(3) TRUE},\ \text{(4) FALSE}$ $\boxed{\text{Answer: (2) and (3)}}$