= Solution
Set $a=\pi/2$ for this part. In the <interval> $(-\pi,\pi)$ the special points are $-a,0,a$. Elsewhere $f_0=\sin(\tan x)$ is a composition of <smooth> <functions>.
At zero, $\tan x=x+O(x^3)$ and therefore $\sin(\tan x)=x+O(x^3)$. Its assigned value is zero, so the <function> is continuous and <differentiable> there, with $f_0'(0)=1$.
At either tangent pole $\pm a$, the bounded <sine> oscillates without a limit. For an explicit check at $a$, let $x_j=\arctan(2\pi j+\pi/2)$ and $u_j=\arctan(2\pi j+3\pi/2)$. Both approach $a$ from below, but the <function> values are respectively $1$ and $-1$. Negating these sequences gives the same obstruction at $-a$. Assigning the value zero at the poles cannot repair either limit.
Thus \b[$f_0$ is discontinuous and nondifferentiable exactly at $x=\pm\pi/2$], and is <smooth> elsewhere in the specified <interval>.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1-oscillation-n0.png]
{title=Undamped tangent-phase oscillations with assigned zero values at both poles and an ordinary differentiable crossing at zero}
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