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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