A partial ceiling?

Relevant wiki: Fractional Part Function

The range of $\{x\}$ is always $0 \leq \{x\} < 1$ . However it is 0 only for an integer. Since integers are not irrational, the co-domain of $\{x\}$ does not include 0 in this case. So, $\left \lceil \{x\} \right \rceil$ is always $\boxed{1}$ .

Relevant wiki: Fractional Part Function

For an irrational number $x$ , $0< \{x\}<1$ , so $\left \lceil \left \{ x \right \} \right \rceil = \boxed1$

The range is always $0 \le \{x\}<1$ and $\lceil x \rceil=\lfloor x \rfloor+1$ so $\lceil \{x\} \rceil=\boxed{\large{1}}$