1. The problem asks for the domain of the step function $f(x) = \lceil 2x \rceil - 1$. 2. The domain of a function is the set of all possible input values ($x$) for which the function is defined. 3. The ceiling function $\lceil y \rceil$ is defined for all real numbers $y$, and it returns the smallest integer greater than or equal to $y$. 4. Since $2x$ is defined for all real numbers $x$, and the ceiling function is defined for all real numbers, the composition $\lceil 2x \rceil$ is also defined for all real numbers. 5. Subtracting 1 from $\lceil 2x \rceil$ does not restrict the domain. 6. Therefore, the domain of $f(x) = \lceil 2x \rceil - 1$ is all real numbers. 7. In set notation, this is written as $\{x \mid x \text{ is a real number}\}$. Final answer: $\boxed{\{x \mid x \text{ is a real number}\}}$