I have this problem: let $X : [0,1] \rightarrow \mathbb{R}$. Find all inverse images of the sets of the form $(-\infty,a] $ for $X(x)=2 \times \mathbf1 _{[0,\frac{1}{2})}(x) +3 \times \mathbf1 _{[\frac{1}{2}, 1]}(x)$
I thought the answer might be (-∞,0) or [2,3] from my working but neither seems to fit the required form (-∞,a]. Please let me know if there is something I have missed in my understanding
Guide:
The question is asking for inverse images of sets of $(-\infty, a]$. The answer need not be of that form.
Note that the range of $X$ takes a few values only, which are $0$, $2$, $3$.
For example, when $a<0$, $f^{-1}(-\infty, a]=\emptyset$
Consider what happens to $f^{-1}(-\infty,a]$ when $0 \le a < 2$, $2 \le a < 3$, and $a \ge 3$ separately.