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Here's where things get really tricky. The function f that I just defined is 1-1 but it is not onto. For example, the number z=1/11 is not in the range of f. In fact, 1/11 = 0.090909... . If this number is in the range of f the we would need to have z=f(x,y), where the decimal expansion of x is given by the digits in the odd-numbered places in the decimal expansion of z, and the decimal expansion of y is given by the digits in the even-numbered places in the decimal expansion of z. In other words, x=0.00000... and y=0.99999... . But neither of these is allowed: x and y are supposed to be in the open unit interval, so they cannot be equal to the endpoints 0 or 1 of the interval. Also, we don't allow expansions ending in an infinite string of 9s. That shows that the function f is not onto.