  In standard rounding, we want to map each value $X$ in a large continuous
space (e.g., $R$) to a nearby point $P$ from a discrete subset (e.g., $Z$).
This process seems to be inherently discontinuous in the sense that two
consecutive noisy measurements $X_1$ and $X_2$ of the same value may be
extremely close to each other and yet they can be rounded to different points
$P_1\ne P_2$, which is undesirable in many applications. In this paper we show
how to make the rounding process perfectly continuous in the sense that it maps
any pair of sufficiently close measurements to the same point. We call such a
process consistent rounding, and make it possible by allowing a small amount of
information about the first measurement $X_1$ to be unidirectionally
communicated to and used by the rounding process of $X_2$.
  The fault tolerance of a consistent rounding scheme is defined by the maximum
distance between pairs of measurements which guarantees that they are always
rounded to the same point, and our goal is to study the possible tradeoffs
between the amount of information provided and the achievable fault tolerance
for various types of spaces. When the measurements $X_i$ are arbitrary vectors
in $R^d$, we show that communicating $\log_2(d+1)$ bits of information is both
sufficient and necessary (in the worst case) in order to achieve consistent
rounding for some positive fault tolerance, and when d=3 we obtain a tight
upper and lower asymptotic bound of $(0.561+o(1))k^{1/3}$ on the achievable
fault tolerance when we reveal $\log_2(k)$ bits of information about how $X_1$
was rounded. We analyze the problem by considering the possible colored tilings
of the space with $k$ available colors, and obtain our upper and lower bounds
with a variety of mathematical techniques including isoperimetric inequalities,
the Brunn-Minkowski theorem, sphere packing bounds, and \v{C}ech cohomology.
