Question

$f\left(x\right)=\frac{|x^3-3x^2+2x|}{x^3-3x^2+2x}.$Find the set of points a, where $\underset{x\to a}{lim}f\left(x\right)$ does not exist. Number of such points is?

Answer 1

$f\left(x\right)=\frac{\left|x(x-1)(x-2)\right|}{x(x-1)(x-2)}=\frac{\left|x\right|}{x}\frac{|x-1|}{x-1}\frac{|x-2|}{x-2}$
Consider the points $x=0,1,2;$
We redefine the function,
$f\left(x\right)=\{\begin{array}{ccc}-1& x<0& (-1)(-1)(-1)=-1\\ 1& 0<x<1& \left(1\right)(-1)(-1)=1\\ -1& 1<x<2& \left(1\right)\left(1\right)(-1)=-1\\ 1& x>2& \left(1\right)\left(1\right)\left(1\right)=1\end{array}$
$limf\left(x\right)$ as $x\to 0,1,2$ does not exist as both R.H.L. and L.H.L. will be differnet.
Hence 0, 1, 2. are points of discontinuity.