I'm trying to find the extremes of the ranges of $x$ in order to find a valid PDF having the following form:
$$ f(x)= \begin{cases} x, \qquad x \in [a, \, b] \\ \frac{1}{x},\qquad x \in [b, \, c] \end{cases} $$
It has simmetry and it's an odd function.
Which could be the values of a, b and c?
$$ \int_a^b x\, dx + \int_b^c \frac{1}{x} \, dx = \frac{b^2} 2 - \frac{a^2} 2 + \log_e c - \log_e b $$ Here you will want $c\ge b\ge a\ge0.$ If the value of $a$ decreases to make $\dfrac{b^2}2-\dfrac{a^2}2$ bigger, but not bigger than $1,$ then you can compensate by making $c$ smaller. You have two degrees of freedom here: once $a$ and $b$ are determined, you need to find the appropriate value of $c.$ You have $$ \log_e c = 1 + \frac{a^2}2 - \frac{b^2} 2 + \log_e b $$ Therefore $$ c = \exp\left( 1 + \frac{a^2}2 - \frac{b^2} 2 + \log_e b \right). $$