Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative$$\frac{4 \delta\left(x - 1\right) - \frac{\delta\left(x\right)}{\sqrt{\left|{x}\right|}} + \frac{\operatorname{sign}^{2}{\left(x \right)}}{4 \left|{x}\right|^{\frac{3}{2}}} + \frac{\left(4 \operatorname{sign}{\left(x - 1 \right)} - \frac{\operatorname{sign}{\left(x \right)}}{\sqrt{\left|{x}\right|}}\right)^{2}}{2 \left(\sqrt{\left|{x}\right|} - 2 \left|{x - 1}\right|\right)}}{\left(\sqrt{\left|{x}\right|} - 2 \left|{x - 1}\right|\right)^{2}} = 0$$
Solve this equationSolutions are not found,
maybe, the function has no inflections