For N ∈ N_≥2 and α ∈ R such that 0 < α ≤ N − 1, the continued fraction map T_α: [α, α+1] → [α, α+1) is defined as T_α (x):= N/x−d(x), where d: [α, α+1] → N is defined by d(x):= ⌊N/x − α⌋. A maximal open interval (a, b) ⊂ I_α is called a gap of I_α if for almost every x ∈ I_α there is an n₀ (x) ∈ N such that x_n /∈ (a, b) for all n ≥ n₀ . In this paper, all conditions are given in which I_α is gapless. For α =^√N − 1 it is shown that the number of gaps is a finite, monotonically non-decreasing and unbounded function of N.