Let $f : [a,b] \rightarrow [1, \infty)$ be a continuous function and let $g : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$g(x) = \begin{cases} 0 & \text{if } x < a \\ \int_{a}^{x} f(t)\, dt & \text{if } a \leq x \leq b \\ \int_{a}^{b} f(t)\, dt & \text{if } x > b \end{cases}$$ Then (A) $g(x)$ is continuous but not differentiable at $a$ (B) $g(x)$ is differentiable on $\mathbb{R}$ (C) $g(x)$ is continuous but not differentiable at $b$ (D) $g(x)$ is continuous and differentiable at either $a$ or $b$ but not both
Let $f : [a,b] \rightarrow [1, \infty)$ be a continuous function and let $g : \mathbb{R} \rightarrow \mathbb{R}$ be defined as
$$g(x) = \begin{cases} 0 & \text{if } x < a \\ \int_{a}^{x} f(t)\, dt & \text{if } a \leq x \leq b \\ \int_{a}^{b} f(t)\, dt & \text{if } x > b \end{cases}$$
Then\\
(A) $g(x)$ is continuous but not differentiable at $a$\\
(B) $g(x)$ is differentiable on $\mathbb{R}$\\
(C) $g(x)$ is continuous but not differentiable at $b$\\
(D) $g(x)$ is continuous and differentiable at either $a$ or $b$ but not both