We use the theory of continued fractions over function fields in the setting of hyperelliptic
curves of equation y^2 = f(x); with deg(f) = 2g+2. By introducing a new sequence
of polynomials defined in terms of the partial quotients of the continued fraction expansion
of y, we are able to bound the sum of the degrees of consecutive partial quotients.
This allows us both (1) to improve the known naive upper bound for the order N of the
divisor at infinity on a hyperelliptic curve; and, (2) to apply a naive method to search for
hyperelliptic curves of given genus g and order N. In particular, we present new families
defined over Q with N = 11 and 1 <g < 11.