apply ContinuousOn.abs ?_ |>.mul ?_ |>.locallyIntegrableOn (by simp)
case hf
r : ℝ
rneg : r < 0
N : ℕ
Npos : 0 < N
⊢ LocallyIntegrableOn (fun x => |x ^ (r - 1)| * |x.log|) (Ici ↑N) volume
r : ℝ
rneg : r < 0
N : ℕ
Npos : 0 < N
⊢ ContinuousOn (fun x => x ^ (r - 1)) (Ici ↑N)

r : ℝ
rneg : r < 0
N : ℕ
Npos : 0 < N
⊢ ContinuousOn (fun x => |x.log|) (Ici ↑N)