Mathematics

Problem

Let

f: [0, \infty) \rightarrow [0, 3]

be a function defined by

f(x) = \begin{cases} \max\{\sin t: 0 \leq t \leq x\}, & 0 \leq x \leq \pi \\ 2 + \cos x, & x > \pi \end{cases}

Then which of the following is true? (a)

f

is continuous everywhere but not differentiable exactly at one point in

(0, \infty)

(b)

f

is differentiable everywhere in

(0, \infty)

(c)

f

is not continuous exactly at two points in

(0, \infty)

(d)

f

is continuous everywhere but not differentiable exactly at two points in

(0, \infty)