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

A

g(x) is continuous but not differentiable at a

B

g(x) is differentiable on 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