A function $f(x)$ is given below. Which of the following is true for $f(x)$?

$f(x) = \left\{ {\begin{array}{*{20}{c}}
x&{x < 0}\\\
0&{x = 0}\\\
{\sin x}&{0 < x < \frac{\pi }{2}}\\\
1&{x \ge \frac{\pi }{2}}
\end{array}} \right.$

I.$f(x)$ is continuous and differentiable at $x=0$.II.$f(x)$ is differentiable at $x=\cfrac {\pi}{2}$.III.$f(x)$ is continuous at $x = 0$ and at $x=\cfrac {\pi}{2}$.

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