...b]. Let E be a normed vector space and V a functional defined in E. V is continuous at a point y0 e E if for every e > 0 there is a <5 > 0 such that \V(y)-V(y0)\ < e whenever \\y-y0\\ < 6. A functional V is said to be linear if it is continuous and...

...should be proved by the student. 27.2 / Definition The funetion f:A — *(R is uniformly continuous if for every e > 0 there is a 5 > 0 such that if x, y € A and d(x, y) < 5, thend(/(x),/(y)) < t. Clearly every uniformly continuous funetion is...

...condition holds for the Henstock integral. Lemma 3.4. A function f is Henstock integrable on [a,b] if and only if for every e > 0 there is a 5(£) > 0 such that for any S— fine divisions D - ([u,v];£) and D' - ([u'.v'];£'> we have |£ f(O(vu) - I f(?')(v'-u')|...

...integrable on [a,b]. Definition l7.l. A measurable function f defined on [a,b] is said to have LSRS if for every e > 0 there is a 5(£) > 0 such that for every te [a,b] we have |£f(O(vu)| < c whenever D - ([u,v];£) is a 5-fine division of an interval...

...according as re R+ or re R+. Finally, we say that a function/(-): X-» R" is uniformly continuous on X if for every e > 0 there is a <5 > 0 such that x.yeX and \\x - y\\ < d imply that || f(x) - f(y)\\ < E. 12.7.1 Proof of T 12.1 Since F( • ) is a...

...integral $RAdxi dx2 . . . dxk of a fc-form over a /^-dimensional rectangle is said to be convergent if for every e > 0 there is a 5 > 0 such that |Z(«) — £(«')I < e whenever £(«), £(«') are approximating sums to $RAdxi dx2 . . . dxk in...

...of a fixed (or periodic) point. A fixed point x in the state space R" is said to be Lyapunov-stable if, for every e > 0, there is a <5 > 0 such that all points within distance ô of x have trajectories that remain within £ of a:. A final notion is...

...xe X , we say T is continuous. One verifies, just as for real functions, that T is continuous at x if and only if, for every e > 0, there is a <5 > 0 so that a(T(x), T(y)) < e, whenever p(x, y) < 5. Also T is continuous at every point in X if and only...

...function whose domain contains E. We say that F is absolutely continuous in the restricted sense on E if, for every e > 0 there is a (5 > 0 such that for every sequence {[an,bn]} of non-overlapping intervals whose endpoints belong to E, ^2n(bn — an) <...

...a subset of R"1 . Let h : A -> B be a function. We say that h is continuous if, for every ae A and for every e > 0, there is a <5 > 0, such that for every x 6 A if \\x — a\\ < 8 then \\h(x) - h(a)\\ <£. This definition is probably familiar to the...