If graphical observations of a limit are feasible for totient approxinmation\extension onto $\mathbb R$

Question

Essentially I want to know if the following can be considered true, despite the fact that the Euler totient is not actually a continuous function on $\mathbb R$ for which all the implications of an integral would apply:

$$\int_0^{\infty}\Bigl\lfloor \frac{n\, \varphi(n)}{n - 1}\Bigr\rfloor\,\operatorname{d}n-\int_0^{\infty}n\,\operatorname{d}n=0$$