Connection between ceil and round
I have two rounding functions.
The first is `ceil`, which always finds the smallest integer that is at least as large as the input. E.g. `ceil(0) = 0`, `ceil(0.1) = 1`, `ceil(0.5) = 1`, `ceil(0.9) = 1`.
The second is `round`, which always finds the integer closest to the input. For the middle (.5), it is defined as returning the next larger integer. E.g. `round(0) = 0`, `round(0.1) = 0`, `round(0.5) = 1`, `round(0.9) = 1`.
Is there a connection between them that allows to substitute `ceil` by `round`?
Like `ceil(x) = g(round(f(x)))` for some functions `f`, `g` that do not contain `ceil`? Do such functions exist?
I only care about nonnegative numbers as input for `ceil` and `round`.
With the given rounding rule for half integers, your best bet is