# Let's do some calculus! (7)

Calculus Level 3

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 $$b$$.