Let, e.g.
$$ f(x) = \begin{cases} x,\quad x<1, \\ 1,\quad x\geq1, \end{cases} $$
a piecewise linear function. Does the following hold for $g(x) = 1/x$?
$$ \begin{align} g(f(x)) &\stackrel{?}{=} \begin{cases} g(x), &x<1, \\ g(1), &x\geq1, \end{cases} \\ &\stackrel{?}{=} \begin{cases} 1/x, &x<1, \\ 1, &x\geq1. \end{cases} \end{align} $$
And congruently, with $x_1<1\wedge x_2>1$, does the following hold as well?
$$ \int_{x_1}^{x_2} \frac{\text{d}x}{f(x)} \stackrel{?}{=} \int_{x_1}^{x_2} g(f(x))\,\text{d}x \stackrel{?}{=} \int_{x_1}^1 \frac{\text{d}x}{x} + \int_1^{x_2} \frac{\text{d}x}{1}. $$