questions including [x]

Every real number can be placed between two consecutive integers, such that \(k\leq{x}<k+1\).
For every given \(x\), the integer \(k\)with the above property is unique. By definition, then \([x] = k\).
You can always write \([x]\leq{x}<[x]+1.\)
If \(x=5,\) \([5]=5\). If \(x=5.67\), \([x]=5.\)
\([0]=0\), \([0.84]=0\), but \([-0.34]=-1.\)

If \(x\) is an integer, than \([x]\) is simply \(x.\)
For non-integer \(x\), try to visualize the number line. The closest integer to \(x\) from the left is \([x].\)
We used to call it the integer part function. For positive numbers, just ignore the decimal part and leave the integer part of the number.
For negative numbers, don't forget to look at the left of your number. \([-2.7] = -3\) and not \(-2.\)

So, if \([x] = 0\), then certainly \(0\leq{x}<1\), meaning \(x\) can be any number between 0 and 1, 0 inclusive, 1 exclusive.