Q: Do you think there is a continuous function mapping [0,1] onto the unit square?There does exist a continuous function mapping the unit square onto [0,1]:
If we represent the unit square as {(x,y) in R2 | x in [o,1], y in [0,1]
} Then f(x,y) = x is a continuous function mapping the unit square onto [0,1] (can rig. prove via def'n w/ delta=epsilon, tri-inequal) So I personally, didn't think that there was a cont. funct. mapping [0,1] onto the unit square but math always finds a way to amaze me! Apparently, such counter-intuitive functions were discovered a hundred years ago, or so! Fun tangent: the cardinality of a unit interval is the same as the cardinality of any finite-dim manifold... i.e. for example: a square (or even a cube (or hypercube)) and its side contain the same number of points.
http://en.wikipedia.org/wiki/Space-filling_curve
Ok reading too much of this stuff gave me a headache. math overload. knock yourselves out, kids!
