Let g, h be two real-valued convex functions on R. Let m(x) = max{h(x), g(x)). Prove...
4. Let f: X Y +R be any real valued function. Show that max min f(x,y) < min max f(x,y) REX YEY yey reX
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists M> 0 such that n(x) M for all r E [0,1] and all n N. (b) Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? Prove or give a counterexample (2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists...
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R. Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
x+6 For the real-valued functions g(x)= and h(x) = 4x+15, find the composition g h and specify its domain using interval notation. (goh) (x) = 0 号 [P] 。 口口] (CD) 可 口UD (口, 可 口,口) Domain of g h : x 6 ?
For the real-valued functions g(x)=x+2 and h(x) -x-1, find the composition g h and specify its domain using interval notation. De (* •)(=) = 0 Domain of s : 0 x 5 ?
3. Let la, b) on [a, b]. Define | lo, by R and Cla, b be the space of continuously differentiable real-valued functions lsup () sup |f'() rEla,b rela,b Prove that (Cl a) is a Banach space 3. Let la, b) on [a, b]. Define | lo, by R and Cla, b be the space of continuously differentiable real-valued functions lsup () sup |f'() rEla,b rela,b Prove that (Cl a) is a Banach space
Number 6 please S. Let ) be a sequence of continuous real-valued functions that converges uniformly to a function fon a set ECR. Prove that lim S.(z) =S(x) for every sequence (x.) C Esuch that ,E E 6. Let ECRand let D be a dense subset of E. If .) is a sequence of continuous real-valued functions on E. and if () converges unifomly on D. prove that (.) converges uniformly on E. (Recall that D is dense in E...
Let F be the set of all real-valued functions having as domain the set R of all real numbers. Example 2.7 defined the binary operations +- and oon F. In Exercises 29 through 35, either prove the given statement or give a counterexample. 29. Function addition + on F is associative. 30. Function subtraction - on is commutative
Let f.g,h: R + R be functions. Prove that the followings is true or not. If not show an example. a) (g+h) • f = (gºf) + (hof) b) f(g+h) = (fºg) + (f • h)
4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose the sequence {f) converges uniformly on every compact subset of (a, b). Prove thatf is differen- tiable on (a, b) and that f'(x) = lim f(x) for all x E (a, b). 4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose...