= Solution
At $-a$, the prefactor $(x-a)/x$ tends to $2$, so the nonvanishing <amplitude> leaves the oscillatory discontinuity intact. At zero, use $\sin(\tan x)=x+O(x^3)$ to obtain
$$
f_1(x)=\left(1-\frac ax\right)[x+O(x^3)]
=-a+x+O(x^2).
$$
The limit is $-a$, whereas the assigned value is zero. This is a <removable discontinuity>, not a continuous crossing.
At $a$, write $x=a+h$. The estimate
$$
|f_1(a+h)|\leq\frac{|h|}{|a+h|}\longrightarrow0
$$
proves <continuity> at the assigned zero. But its <difference quotient> is
$$
\frac{f_1(a+h)-f_1(a)}h=\frac{\sin(\tan(a+h))}{a+h}.
$$
The sequences from part (i) make this approach $1/a$ and $-1/a$, so the <derivative> does not exist. This illustrates the distinct <continuity and differentiability thresholds for a damped oscillation>.
Therefore \b[the discontinuities are $x=-\pi/2$ and $x=0$; the nondifferentiability points are $x=-\pi/2,0,\pi/2$]. The <function> is <smooth> elsewhere.
\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-n1.png]
{title=Linear suppression of the positive-pole oscillations, with a removable discontinuity at zero}
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