Let $f : [a, b] \to [1, \infty)$ be a continuous function and let $g : R \to R$ be defined as $g(x) = \begin{cases} 0, & \text{if } x < a, \\ \int_{a}^{x} f(t)\,dt, & \text{if } a \le x \le b; \\ \int_{a}^{b} f(t)\,dt, & \text{if } x > b. \end{cases}$ then

Let f : [a, b] → [1, ∞) be a continuous function and let g : R → R be defined as g(x) = 0, if x < a, ∫ₐˣ f(t) dt, if a ≤ x ≤ b; ∫ₐᵇ f(t) dt, if x > b. then

$\text{Given: } f : [a, b] \to [1, \infty) \text{ continuous},\quad g : \mathbb{R} \to \mathbb{R}$

$g(x) = 0\ (x < a);\ \int_a^x f\,dt\ (a \le x \le b);\ \int_a^b f\,dt\ (x > b)$

$\text{To find: which statement (1)--(4) is correct}$

$\text{FTC}: g \text{ is continuous on } \mathbb{R}$

$g(a^-) = 0 = g(a); \quad g(b^-) = g(b) = g(b^+) = \int_a^b f$

$g'(a^-) = \dfrac{d}{dx}(0) = 0, \qquad g'(a^+) = f(a) \ge 1$

$0 \ne f(a) \Rightarrow g \text{ not differentiable at } a \Rightarrow (1)\ \text{TRUE}$

$g'(b^-) = f(b) \ge 1, \qquad g'(b^+) = \dfrac{d}{dx}\left(\int_a^b f\right) = 0$

$f(b) \ne 0 \Rightarrow g \text{ not differentiable at } b \Rightarrow (3)\ \text{TRUE}$

$(2)\ \text{false (not differentiable at } a \text{ and } b)$

$(4)\ \text{false (not differentiable at either)}$

$\text{TRUE: } (1)\ \text{and}\ (3)$

$\boxed{(1)\ \text{and}\ (3)}$