Sentence examples for in view of the continuity from inspiring English sources

Note that the operator P is continuous in view of the continuity of f.

Further (mathcal{S}_{1}) is continuous in view of the continuity of f.

Observe that T is continuous in view of the continuity of f, I k and I k *.

Firstly, the operator T is continuous in view of the continuity of functions (f t,u(t))) and (G t,s)).

Proof It is easy to see that the operator F is continuous in view of the continuity of G and f.

The operator (A Pto P) is continuous in view of the continuity of (G t,s)) and (f t,u(t))).

In view of the Hölder continuity of u, i.e., (3.48), Arzelà-Ascoli theorem asserts ((u(cdot,t))_{t>1}) is relatively compact in (C^{0}(bar{Omega})).

In view of the absolute continuity of the Lebesgue integral and properties (c) and (f), for any (epsilon>0), we can choose a number (k'inmathbf{N}) large enough such that |l|_{L^{2} omega_{t^backslashomega_{t'})}< sqrt{epsilon}, qquad| tilde{u}|_{L^{2} omega_{t^backslashomega_{t'})}< sqrt{ epsilon}.

In view of the absolute continuity of the Lebesgue integral and the properties (b) and (f), for any (epsilon>0) we can choose a number (k'in mathbf{N}) large enough such that |l|_{L^{2} omega_{t^backslashomega_{t'})}< sqrt{epsilon}, qquad | tilde{u}|_{L^{2} omega_{t^backslashomega_{t'})}< sqrt{epsilon}.

Then we have p(fx,x) leq p(fx,x_{n+1}) + p(x_{n+1},x), and in view of the 0-continuity of f at x, we also obtain (lim_{n rightarrowinfty}p(fx,x_{n+1}) = 0), that is, (p(fx,x) = 0).

Proof The operator T : K → X is continuous in view of the nonnegativity and continuity of the functions G ( t, s ) and f ( t, u ( t ), D 0 + β u ( t ) ).