Let $f:[0,\infty ) \to [0,3]$ be a function defined by $$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi } \cr } } \right.$$Then which of the following is true?

Question

Let $f:[0,\infty ) \to [0,3]$ be a function defined by

$$f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi } \cr } } \right.$$

Then which of the following is true?

f is continuous everywhere but not differentiable exactly at one point in (0, $\infty$)

f is differentiable everywhere in (0, $\infty$)

f is not continuous exactly at two points in (0, $\infty$)

f is continuous everywhere but not differentiable exactly at two points in (0, $\infty$)

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