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Real Analysis

Real Analysis : Limit Superior

Let a_n be bounded sequence.prove that a-the sequence defined by y_n=sup{a_k:k>=n} converges. b- Prove that lim inf a_n<=lim sup a_n for every bounded sequence and give example of a sequence which the inequality is strict.

Real Analysis: Differentiability and Limits

Prove : Assume f and g are continous functions defined on interval contaning a, and assume that f and g are differentiable on tis interval with the possible exception of the point a. If f(a)=0 and g(a)=0 then lim f'(x)/g'(x)=L as x->a implies lim f(x)/g(x)=L as x->a.

Real Analysis : Neighborhoods

Assume g:(a,b)->R is differentiable at some point c belong to (a,b). If g'(c)not= 0 show that there exists a delta neighborhood V_delta (c) subset or equal to (a,b) for which g(x) not= g(c) for all x belong to V_delta (c).

Real Analysis : Differentiable and Increasing Functions

A-a function f:(a,b)->R is increasing on (a,b) if f(x)<=f(y) whenever x<y in (a,b). Assume f is differentiable on (a,b). Show that f is increasing on (a,b)if and only if f'(x)>=0 for all x belong to (a,b). b-show that the function g(x){x/(2+x^2 sin(1/x)) if x not=0 0 if x=0 is differentiable on R and satisfies g'(0)>0.Now

Real Analysis : Twice Differentiable Functions

Let g:[0,1]->R be twice-differentiable (i.e both g and g' are differentiable functions) with g''(x)>0 for all x belong to [0,1].if g(0)>0 and g(1)=1 show that g(d)=d for some point d belong to (0,1) if and only if g'(1)>1.

Real Analysis: Points on a Differentiable Function

Let h be a differentiable function defined on the interval [0,3], and assume that h(0)=1 h(1)=2 and h(3)=2. a- argue that there exists a point d belong to [0,3] where h(d)=d. b-argue that at some point c we have h'(c)=1/3. c-argue that h'(x)=1/4 at some point in the domain.

Real Analysis : Contractiveness

Prove that a function f is contractive on a set A if there exists a constant 0<s<1 such that Absolute value of f(x)-f(y)<=s*Absolute value of x-y for all x,y belong to A.show that if f is differentiable and f' is continous and satisfies Absolute value of f'(x)<1 on a closed interval then f is contractive on this set.

Real analysis : Lipschitz Criterion

A function f:A->R is Lipschitz on A if there exists an M>0 such that Absolute value of f(x)-f(y)/x-y <=M for all x,y belong to A. show that if f is differentiable on a closed interval [a,b] and if f' is continous on [a,b] then f is Lipschtiz on [a,b]. Geomtrically speaking, a function f is Lipschitz if there is a uniform bound

Real Analysis: Differentiability

Prove that if f and g are differentiable functions on an interval A and satisfy f'(x)=g'(x) for all x belong to A, then f(x)=g(x)+k for some constant k belong to R.

Real Analysis : Uniform Convergence

Let (f_n) be a sequence of diffrentiable functions defined on the closed interval [a,b] and assume (f'_n) converges uniformly on [a,b]. Prove that if there exists a point xo belong to [a,b] where f_n(xo) is convergent, then (f_n) converges uniformly on [a,b].

Real Analysis : Cauchy Criterion for Uniform Convergence

Prove that A sequence of functions (f_n) defined on a set A subset or equal to R converges uniformly on A if and only if for every e>0(epsilon) there exists an N belong to N such that Absolute value of f_n (x)-f_m (x)<e for all m,n>=N and all x belong to A.

Real analysis

G(x)=Sum sign(m top n=0 bottom)(1/2^n)h(2^n x).for more inf. please check #30026,#30028,#30029. show that (g(x_m)-g(0))/(x_m - 0)=m+1, and use this to prove that g'(0) does not exist. any temptation to say something like g'(0)=oo should be resisted. setting x_m=-(1/2^m) in the previous argument produces difference heading to

Real analysis

Taking the continuity of h(x) as given in#30026,#30028 by using any of the functional limits and continuity theorems prove that the finite sum g_m (x)=sum sign(oo top n=0 bottom) of 1/2^n h(2^n x) is continous on R

Real Analysis: Jump Discontinuity

Let f:R->R be increasing. Prove that if lim f(x) as x->c^+ and if lim f(x) as x->c^- must each exist at every point c belong to R. Argue that the only type of discontinuity a monotone function can have is a jump discontinuity.

Real Analysis - Discontinuity

Prove that a- if lim f(x) as x->c exists but has a value different from f(c) the discontinuity at c is called removable, b-if lim f(x) as x->c^+ not =lim f(x) as x->c^-, then f has a jump discontinuity at c, c-if lim f(x) as x->c does not exists for some other rea

Real Analysis : Limits

Prove that if f:A->R and a limit point c of A , lim f(x)=L as x->c if and only if lim f(x)=L as x->c^-(left handed limit) and lim f(x)=L as x->c^+(right handed limit).

Real Analysis: Continuous Extension Theorem

A: Show that a uniformly continous function preserves Cauchy sequences; that is, if f:A->R is uniformly continous and (x_n) subset or equal of A is a Cauchy sequence then show f(x_n) is a Cauchy sequence. B: Let g be a continous function on the open interval (a,b). prove that g is uniformly continous on (a,b) if and only if i

Uniformly Continuous Problems

A-Assume that f:[0,oo)->R is continous at every point in its domain.show that if there exists b>0 such that f is uniformly continous on the set[b,oo), then f is uniformly continous on [0,oo). b-Prove that f(x)=sqrt[x] is uniformly contionus on [0,oo).

Real Analysis : Uniformly Continuous

Assume that g is defined on an open interval (a,c) and it is known to be uniformly contionus on (a,b] and [b,c) where a<b<c.prove that g is uniformly continous on (a,c).

Real analysis: Existence Of A Fixed Point

Let f be a continuous function on the closed interval [0,1] with range also contained in [0,1].Prove that f must have a fixed point; that is, show f(x)=x for at least one value of x belong to [0,1].

Find the Laurent

Find the Laurent series about all singular points of f(z) = 1/(z(z+1)^2) {See attachment}.

Proof Regarding Continuous Functions

Let f be a function defined on all of R that satisfies the additive condition f(x+y)=f(x)+f(y) for all x,y belong to R a- Show that f(0)=0 and that f(-x)=-f(x) for all x belong to R. b- Show that if f is continuous at x=0 then f is continuous at every point in R c- Let k=f(1) show that f f(n)=kn for all n belong to N and

Proof Regarding Continuity and Contraction Mapping

(contraction mapping theorem).let f be a function defined on all of R and assume there is a constant c such that 0<c<1 and Absolute value of f(x)-f(y)<= c Absolute value of x-y for all x,y belong to R show that f is continuous on R.

Real Analysis

A- Show that if a function is continuous on all of R and equal to 0 at every rational point then it must be identically 0 on all of R b- if f and g are continuous on all of R and f(r)=g(r) at every rational point,must f and g be the same function?