On the notion of a 'continuous' decoration: I believe
we have to be careful with its definition. While it is
possible that the underlying function on the Reals can
be flagged continuous or discontinuous as the case may
be, we are really defining functions on a finite set of
points (or intervals) that have a lower bound on just
how dense they can be (according to our working
precision).
Thus we can have examples of both continuous &
discontinuous functions that appear to be the opposite
when realized in floating-point.
For example, the sine function is perfectly continuous
over all of the Reals but as soon as the interval values
exceed a few times b^p the relationship between the sines
of consecutive representable numbers can take on a random
& discontinuous appearence. Then, it can falsely appear
continuous again for some b^q for q > p if b^q is close
to some large multiple of 2pi. Indeed, from something
like 2*pi*b^p onward, the sine of any non-point interval
will be [-1,1] & the sine of any point interval has pretty
much no meaning WRT a convergence method of any kind.
So while we may define 'continuous' or even NEED to
define it, we may not recognise it when we see it unless
the definition is very carefully crafted.