=+7.13. The functional equation (7.30) and the assumption that Q is bounded suffice to determine Q completely. First, Q(0) and Q(1) must be 0 and 1, respectively, and so (7.31) holds. Let Tox = {x and T,x = ; x + ;; let fox = px and fix =p + qx.

Then Q(TL, . T. x) =f ... ... f ., Q(x). If the binary expansions of x and y both begin with the digits u1 ...., u ,,, they have the form x = T. ... Tx' and y = T .. ... Ty'. If K bounds Q and if m = max( p, q), it follows that

[Q(x)-Q(y)| ≤ Km". Therefore, Q is continuous and satisfies (7.31) and (7.33).