Suppose b k {b_k} denotes either ϕ ( k ) \phi (k) or ϕ ( p k ) ( k = 1 , 2 , … ) \phi ({p_k})\;(k = 1,2, \ldots ) where the polynomial ϕ \phi maps N \mathbb {N} to N \mathbb {N} and p k {p_k} denotes the k k th rational prime. Suppose ( c k ( x ) ) k = 1 ∞ ({c_k}(x))_{k = 1}^\infty denotes the sequences of partial quotients of the continued function expansion of the real number x x . Then for certain functions F : R ⩾ 0 → R F:{\mathbb {R}_{ \geqslant 0}} \to \mathbb {R} we show that \[ lim N → ∞ F − 1 [ F ( c b 1 ( x ) ) + ⋯ + F ( c b k ( x ) ) N ] = F − 1 [ 1 ( log ⁡ 2 ) ∫ 0 1 F ( c 1 ( x ) ) 1 + x d x ] \lim \limits _{N \to \infty } {F^{ - 1}}\left [ {\frac {{F({c_{{b_1}}}(x)) + \cdots + F({c_{{b_k}}}(x))}} {N}} \right ] = {F^{ - 1}}\left [ {\frac {1} {{(\log 2)}}\int _0^1 {\frac {{F({c_1}(x))}} {{1 + x}}dx} } \right ] \] almost everywhere with respect to Lebesgue measure. This result with b k = k {b_k} = k is classical and due to Ryll-Nardzewski.