How to round 0.4999... ? Is it 0 or 1? If you want to round the repeating decimal 0.4999... to a whole number what is the right answer? Is it 0 or 1?

On the one hand you only look at the first digit when you round numbers which in this case is 4 so the answer should be 0.

On the other hand 0.4999... is only another representation for 0.5 which makes the result 1.

My question
Which of these rules wins? (My gut feeling is that the most consistent result should be 1)

Edit
Just for clarification: What is meant by rounding here is replacing a fractional decimal number by one with fewer digits with the "Round half up"-Tie-breaking method.
It's $1$, because $0.49\ldots$ is the same as $0.5$. If rounding is to be well-defined, it can't map one real number to two integers, so whatever it maps $0.49\ldots$ to, it better maps it to the same integer as $0.5$. You could round both to $0$, of course, but that wouldn't then be the way we usually round.

What this shows you is that rounding doesn't commute with limits, i.e. there's a difference between find the limit of a sequence and then rounding, and rounding first and then finding the limit. As you correctly observed, all values $a_n = 0.4\underbrace{9\ldots 9}_{n\text{ times}}$ are rounded down to zero. Thus, $$
  \lim_{n\to\infty} \text{round }(a_n) = 0
$$
On the other hand, $\lim_{n\to\infty} a_n = 0.5$, and thus $$
  \text{round }\left(\lim_{n\to\infty} a_n\right) = 1
$$

There's another word for functions which don't commute with limit - they're called non-continuous. So what you have discovered is simply that rounding is not a continuous function.
If we round c to d, then c approximately equals d.  If d also consists of an integer, then c lies closer to d than any other integer i.  In other words for all integers i such that i does not equal d, ABS(ci-)>ABS(cd-), where ABS(ci-) indicates the number by which c and i differ by (the absolute value of the difference of c an i).  This implies that we can use the distance metric between real numbers to see how rounding works as follows.

Consider the set S=={[l, lu+2/), (lu+2/, u]} where ul-=1.  In other words, l and u differ by 1, and we consider the set of all numbers between x and y except for the midpoint m==lu+2/ between them.  We can use the metric function D(x,y)=|xy-|=ABS(xy-), where x belongs to S and y belongs to {l, u}, to determine how to round all members of S by following rule:

If D(x, l)>D(x, u), then round to u.
If D(x, u)>D(x, l), then round to l.

Since .5=.49999..., it follows that D(.49999..., 1)=D(.49999...., 0)=.4999...=.5.  Thus, the distance function D(x, y)=ABS(xy-) does not give us any guide as to how to round here.

Consequently, we have several different choices that we can make which will not go against our intuition of rounding as follows:


Choose to round .49999... to 1 (maybe you value larger numbers in general),  
Choose to round .49999... to 0 (maybe you value smaller numbers in general),
Choose to allow .49999... to get rounded to both as one pleases or sees fit at particular points in time, or
Choose to round .49999... to neither 0 nor 1 (maybe you don't want to make a choice here).


None of those possibilities win or lose until we have a standard to measure a winner or loser for the rules.  There also doesn't exist a "right answer" here unless we have a means to determine what properties a "right answer" has in this context.  I suspect that many people would resist 3. with the possibility that one could round .4999... to 1 at one time, and then round .4999... to 0 at another as one pleases, thus indicating arbitrariness.  But, it does not follow that such comes as an improper procedure unless one first rates such "arbitrariness" as bad, and something so utterly awful and terrible that we or others must never, ever engage in such "arbitrariness".  Perhaps many people would see 4. as "evasive", but unless there exists a logical basis to picking either 0 or 1, anyone accused of "evasiveness" by picking 4. could very well respond to such a charge, that anyone who sees choosing 4. as indicating "evasiveness", insists dogmatically that all such questions absolutely must have answers, and/or that the people claiming "evasiveness" have copped out of making choices on the basis of logic as much as possible. 

So, what sort of choice do you want to make?