We know that a random variable (RV), say $X$, maps each member of the sample space to the set of real numbers $\mathbb{R}$.
Also, we know that based on the definition of the RV, different sets of points can be selected in $\mathbb{R}$. For example, in tossing a coin, we can define the RV as $X(head)=0$ and $X(tail)=1$. Alternatively, we can define the RV like this $X(head)=-2$ and $X(tail)=2$, so we can define $X$ in many different ways, and based on these different definitions, different sets of points in the set $\mathbb{R}$ are selected.
1) The above-mentioned explanation is right for discontinues RVs, is it correct when we deal with continues ones?
2) if so, then we can change the interval of the continuous RV $X$ on $\mathbb{R}$ based on the definition of $X$. The question is if we change the definition of $X$ which changes the interval of $X$ on $\mathbb{R}$, does the PDF of $X$ also change? and if the answer is yes then we can have many different PDFs for a RV.