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You will need many discrete bits. The number goes up because of the high number of amplitude levels.
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A discrete random variable (RV) can only take a selected number of values whereas a continuous rv can take infinitely many.
No. There are many common functions which are not discrete but the are not continuous everywhere. For example, 1/x is not continuous at x = 0 (it is not even defined there. Then there are curves with step jumps.
No. There are many other distributions, including discrete ones, that are symmetrical.