Sentence examples for I continuous on from inspiring English sources

For each t ∈ I, let B i ( t ) be a positive semidefinite n × n matrix with B i continuous on I, i = 1, 2, …, p and the symbol T denotes the transposition.

where Ω ⊂ ℝ N (N ≥ 3) is a bounded domain with smooth boundary, f ∈ L ∞, p i continuous on Ω ¯ such that 1 < p i (x) < N and ∑ i = 1 N 1 p i - > 1 for all x ∈ Ω ¯, and all i ∈ ℤ[1, N], where p i - : = ess inf x ∈ Ω p i ( x ).

If f : I → R + is a differentiable function in the interior of I, continuous on I, positive and log-convex, then F n 2 ( x ) = ∑ 1 ≤ i < j ≤ n f ( x i ) f ( x j ), x i, x j ∈ Ω, is Schur-convex on I n.

If f : I → R + is a differentiable function in the interior of I, continuous on I, positive and log-convex, then F n n − 1 ( x ) = ∑ 1 ≤ i 1 ⋯ < i n − 1 ≤ n ∏ j = 1 n − 1 f ( x i j ) is Schur-convex on I n.

Then, from the above discussion we know that ( p ) n i is continuous on J for every i ∈ N and p n i ( t ) ≤ p ( t ) ; p n i ( t ) → p ( t ), ∀ t ∈ ( 0, 1 ) as i → + 1.

Remark 3.5 (i) It is easy to see that if M i is continuous on C, then M i is hemicontinuous and bounded on any line segment of C. (ii) It is easy to see that Theorem 3.10 is true if for each i ∈ N, M i : C → H is a κ i -inverse strongly monotone mapping or is a continuous strongly monotone mapping.

ψ ( ⋅, i ) is uniformly continuous in O ¯, and φ ( ⋅, i ) is continuous on ∂O, both for each i ∈ M.  .

φ ( ⋅, i ) is continuous on O ¯, ϕ ( ⋅, ⋅, i ) is continuous on ∂ O × [ 0, T ], for each i ∈ M, where ∂O denotes the boundary of O,   5.

(i) is continuous on ; (ii) for all and there exists a constant for any such that.

In what follows, we prove that V i is continuous on Ω r 0. Fix ε > 0 and take x, y ∈ Ω r 0 such that ∥ x − y ∥ ≤ ε.

Theorem 9 If f is I-ward continuous on a subset E of R, then it is I-sequentially continuous on E. Proof Although the following proof is similar to that of [59], Theorem 1, and that of [7], Theorem 3.2, we give it for completeness.