Interval in which {x} is monotonically increasing function, where { } is fractional part function -
[0,1)
[1,2)
When we look at the graph of fractional part function, it goes from [0, 1) in the interval
[I , I+1), where I is an integer. Thus it behaves as a monotonic function in the interval [I , I+1).
Hence, [0,1) & [1,2) and all other intervals like these are the intervals where {x} is monotonically increasing function.