$\text{⨍(x) =}\left[\mathrm{x}-5\right]\text{+}\left[5-\mathrm{x}\right]$ here $\left[\mathrm{x}\right]$ denotes greatest integer function, is continuous on

$f\left(x\right)=[x-5]+[5-x]$ is continuous on $\mathrm{ℝ}-\mathrm{\mathbb{Z}}$

Concept: The greatest integer function is one that yields an integer that is closest to the specified real value. Additionally known as the step function. The closest integer is used to round off the input using the greatest integer function. Thus, the equation to determine the largest integer is fairly straightforward.

$$\begin{gathered} f\left(x\right)=[x-5]+[5-x] \\ f\left(x\right)=[x-5]+[-(x-5\left)\right] \end{gathered}$$
When (x-5) is an integer then f(x)=0
When (x-5) is not an integer then f(x)= -1
Thus, f(x) is continuous on $\mathrm{ℝ}-\mathrm{\mathbb{Z}}$
Hence, the correct answer is $\mathrm{ℝ}-\mathrm{\mathbb{Z}}$