  The relation between continuous functions and random vectors is revealed in
the paper that the main meaning is described as, for any given continuous
function, there must be a sequence of probability spaces and a sequence of
random vectors where every random vector is defined on one of these probability
spaces, such that the sequence of conditional mathematical expectations formed
by the random vectors uniformly converges to the continuous function. This is
random vector representation of continuous functions, which is regarded as a
bridge to be set up between real function theory and probability theory. By
means of this conclusion, an interesting result about function approximation
theory can be got. The random vectors representation of continuous functionsis
is of important applications in physics. Based on the conclusion, if a large
proportion of certainty phenomena can be described by continuous functions and
random phenomenon can also be described by random variables or vectors, then
any certainty phenomenon must be the limit state of a sequence of random
phenomena. And then, in the approximation from a sequence of random vectors to
a continuous function, the base functions are appropriately selected by us, an
important conclusion for quantum mechanics is deduced: classical mechanics and
quantum mechanics are unified. Particularly, an interesting and very important
conclusion is introduced as the fact that the mass point motion of a
macroscopical object possesses a kind of wave characteristic curve, which is
called wave-mass-point duality.
