Let's do some calculus! (7)

Let $f: [a,b]\longrightarrow[1,\infty)$ be a continuous function and let $g: \mathbb{R} \longrightarrow \mathbb{R}$ be defined as:

$g(x) = \begin{cases} 0 & \text{if} & x<a \\ \\ \displaystyle \int_{a}^{x}{f(t)} \,dt & \text{if} & a\le x \le b \\ \\ \displaystyle \int_{a}^{b}{f(t)} \,dt & \text{if} & x>b \end{cases}$

Then which of the statements is true?

(A) $\ g(x)$ is continuous and differentiable at $b$ .

(B) $\ g(x)$ is continuous and differentiable at $a$ .

(C) $\ g(x)$ is differentiable on $\mathbb R$

(D) $\ g(x)$ is continuous but not differentiable at both $a$ and $b$ .

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