Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 20 / a / ii / 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 obtainThe 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 estimateproves continuity at the assigned zero. But its difference quotient isThe 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 the discontinuities are $x=-\pi /2$ and $x=0$; the nondifferentiability points are $x=-\pi /2,0,\pi /2$. The function is smooth elsewhere.