Jose wrote:
The zero probability event can't occur.

True. But the zero probability items are not not there. One half is
definately there, right between zero and one. The probability of
hitting it is zero.

Your last sentence is wrong. Consider this question: for what value of N
does 1/N = 0? (Hint: "infinity" is not a number you can apply numerical
operations on and expect valid results so you can't say "infinity",
infinity is a concept whereas 1, 2, 3.14, etc. are all values you can use
for N.).

If you insist on considering infinity a number, consider the problems you
encounter - for example, what does infinity/2 evaluate to? How about
infinity/3? Or infinity/billion? Or infinity/infinity?

You are probably being confused by the statement often used in pre-calculus
that:

limit 1/N - 0 as N - infinity. The ratio 1/N is never zero, it "merely"
tends to zero for ever larger values on N.

So the probability of picking 0.5 is infinitesimally small, as Rob points
out, but not zero. It is a mathematically important distinction.