Differentiability of the Minkowski question mark function

We get necessary and sufficient conditions for the derivative of the Minkowski question mark function ? ( x ) to be equal to zero or infinity. These conditions are formulated in terms of sums S x ( t ) = a 1 + ⋯ + a t of partial quotients of continued fraction expansion to x = [ 0 ; a 1 , … , a t ] . In particular we prove that if there exists C such that S x ( t ) ⩽ κ 1 t + log t log 2 + C with κ 1 = 2 log 1 + 5 2 log 2 = 1.38 8 + , then ? ′ ( x ) exists and ? ′ ( x ) = + ∞ . Another result is as follows. Assume that there exists a constant C such that S x ( t ) ⩾ κ 2 t − C with κ 2 = 4 log 5 + 29 2 − 5 log ( 2 + 5 ) log 5 + 29 2 − log ( 2 + 5 ) − log 2 = 4.40 1 + . Then ? ′ ( x ) exists and ? ′ ( x ) = 0 . We show that our conditions on the sum S x ( t ) are the best possible. Our results improve upon earlier theorems by Paradis, Viader, Bibiloni and Dushistova, Moshchevitin.