If a is the number of points of continuity of $f\left(x\right)=\left\{\begin{array}{l}x-1 x \mathrm{is} \mathrm{rational}\\ x^2-x-2 x \mathrm{is} \mathrm{irrational}\end{array}\right.,$ $b$ is the number of
points of discontinuity of $f\left(x\right)=\mathrm{sgn}(x^3-3x+1),c$ is the number of points of non–differentiability of $f\left(x\right)=(\log x)|x^2-4x+3|+2(x-2{)}^{1/3}$ and $d$ is the number of points where the graph of $f\left(x\right)=(\log x)|x^2-4x+3|+2(x-2{)}^{1/3}$ has a sharp turn then the value of $a + b + c + d$ is _____.

$f\left(x\right)=\{\begin{array}{ll}x-1,& x\text{is rational}\\ x^2-x-2,& x\text{is irrational}\end{array}$

is continuous when $x-1=x^2-x-2 \text{or }{x}^2-2x-1=0$ or $x=\frac{2\pm \sqrt{8}}{2}$
Hence, $f\left(x\right)$ is continuous at two points$\Rightarrow a=2$.
$f\left(x\right)=\mathrm{sgn}(x^3-3x+1)$ is discontinuous
when $x^3-3x+1=0$
Now, $y=x^3-3x+1$ has three distinct roots as
$\frac{dy}{dx}=3x^2-3=0\Rightarrow x\pm 1$
Also $f\left(1\right)=1-3=-1 \text{and }f(-1)=-1+3+1=3.$

Hence, the graph of $y=x^3-3x+1$ is
Hence, $f\left(x\right)=\mathrm{sgn}(x^3-3x+1)$ is discontinuous at three points $\Rightarrow b=3$
$f\left(x\right)=(\log x)|x^2-4x+3|+2(x-2{)}^{1/3}$

$=(\log x)|x-1||x-3|+2(x-2{)}^{1/3}$
Which is non-differentiable at $x = 3$and $x = 2$ as at $x = 3$, f(x) has a sharp turn and at $x = 2, f\left(x\right)$ have a vertical tangent. At $x = 1, f\left(x\right)$ is differentiable.
So $c=2 \text{and }d=1$