apply RCLike.continuous_ofReal.continuousOn.comp ?_ (mapsTo_image _ _)
N : ℕ
Npos : 0 < N
s : ℂ
s_re_gt : 0 < s.re
⊢ ContinuousOn (fun x => ↑x.log) (Ioi ↑N)
N : ℕ
Npos : 0 < N
s : ℂ
s_re_gt : 0 < s.re
⊢ ContinuousOn
    (fun x =>
      Decidable.rec (fun h => (fun hx => Real.expOrderIso.symm ⟨|x|, ⋯⟩) h) (fun h => (fun hx => 0) h)
        (x.decidableEq 0))
    (Ioi ↑N)