Fixed Points

Write the address of the point x as a1a2a3... .

Note x is a fixed point of f0 means f0(x) = x.

From the effect of f0 on addresses we see f0(x) = x means

0a1a2a3... = a1a2a3...

Consequently, all the ai = 0 and so x = 0.

The same result is easy to obtain algebraically: x = f0(x) = x/2, so x = 0.

Similarly, x is a fixed point of f1 means f1(x) = x, hence

1a1a2a3... = a1a2a3...

Consequently, all the ai = 1 and so x = 1/2 + 1/4 + 1/8 + ... = 1.

The same result is easy to obtain algebraically: x = f1(x) = x/2 + 1/2, so x/2 = 1/2 and x = 1.

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