I'm looking into communication complexity with real numbers. One problem if we want to define this is that one can encode many real numbers $0.a_1a_2a_3... , 0.b_1b_2b_3..., 0.c_1c_2c_3...$ using only one number $0.a_1b_1c_1a_2b_2c_2...$To get around this problem, existing papers that deal with this issue (such as Abelson (1978), Luo and Tsitiklis, and Chen(1994)) assume that the messages that can be sent between Alice and Bob must be continuously differentiable functions of the inputs $x_1,...,x_n$. Do we need differentiability? Is there any problem if the messages are assumed to be continuous (not necessarily differentiable) functions?  I know that there's no continuous bijective function $f: \mathbb{R}^n \to \mathbb{R}$ for $n > 1$, so it seems like just assuming continuity should be enough. Thanks!
Communication Complexity with real numbers
communication complexity