Question

does ceiling function of -x= floor function of -x for all real x? How?

Answer 1

lets try the point \(x=2.5,\qquad\small{-x=-2.5}\)

ceiling function of \(-x\)
\[\lceil -x\rceil=\lceil -2.5\rceil=-2\]whereas

floor function of \(-x\)
\[\lfloor -x\rfloor=\lfloor -2.5\rfloor=-3\]

Answer 2

so we can see that there exists
at least one point in the real numbers
for which
\[\lceil-x\rceil ≠ \lfloor-x\rfloor\]