Greatest integer of 1-x

Question

In interval x lying between -1/2 to +1/2, what will be the greatest integer of 1-x? The answer is -[x] ,i.e. negative of greatest integer of x but how?

Answer 1

I'll use the more modern notation $\lfloor y\rfloor$ to denote the greatest integer less than or equal to $y.$

Your book appears to have a slight error. In the case where $x=0,$ we have $1-x = 1$ and therefore $\lfloor 1-x\rfloor = 1,$ but $-\lfloor x\rfloor = -(0) = 0.$

In all other cases, the claim would be correct:

• If $0<x<\frac12,$ then $0 < 1-x < 1$ and therefore $\lfloor 1-x\rfloor = 0 = -\lfloor x\rfloor.$
• If $-\frac12 < x < 0,$ then $\lfloor x\rfloor = -1$ and $1 < 1-x < 2,$ so $\lfloor 1-x\rfloor = 1 = -\lfloor x\rfloor.$

However, a statement that is true for all of the interval $\left[-\frac12, \frac12\right]$ except at $0$ and is false at $0$ is not a statement that is true for all of the interval $\left[-\frac12, \frac12\right].$