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Real Analysis

Real Analysis : Bounded Sets

Please see the attached file for the fully formatted problem. Let S be a bounded nonempty set and let S^2 = {s^2 : s E S}. Show that sup S^2 = max((sup S)^2, (inf S)^2).

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 : 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 : 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

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 - 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: 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].

Laurent Series

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

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?

Real Analysis

Assume h:R->R is continuous on R and let K={x:h(x)=0}. show that K is a closed set.

Real Analysis

Let g:A->R and assume that f is a bounded function on A subset or equal to R (i.e there exist M>0 satisfying Absolute value of f(x)<=M for all x belong to A). Show that if lim g(x)=0 as x->c, then g(x)f(x)=0 as x->c as well.

Real Analysis : Convergent and Cauchy Sequences - Five Problems

See attached file for all symbols. --- ? For each of the follwing statements decide if it is true or false. Justify your answer by proving, or finding a couter-example. 1) every bounded sequence of real numbers is convergent. 2) Every convergent sequence is monotone. 3) Every monotone and bounded sequence of real number

Real Analysis

Let C be the Cantor set defined C=intersection sign on top inf bottom n=0 C_n.Define g:[0,1]->R by g(x)={1 if x belong to C and 0 if x does not belong to C. a-show that g fails to be continuous at any point c belong to C. b-prove that g is continuous at every point c does not belong to C

Real analysis

See attachment 1) Locate and classify the extrema of the following functions: Justify your answer 2) Let be continuous functions, such that f(0)+g(0)=0 and f(1)+g(1)=0. assume also that f, g are also differentiable for every use Rolle's theorem to show that there exists an , such that .

Taylor Series

Recall the Taylor series Sum(x^n/n!). The same series can be used to define e^z for a complex number z=a+bi. Use the Taylor series to show that exp(iy) = cos(y) + i sin(y) for any real number y. To do this substitute iy into the series and compute several terms. Look for patterns.

Real Analysis : Open Intervals

Show that it is impossible to write R=U(union sign n=1 bottom, infinity top)F_n where for each n belong to N, F_n is closed set containing no nonempty open intervals.

Real Analysis : Connectedness and Convergent Sequence

Show that A set E subset or equal to R is connected if and only if, for all nonempty disjoint sets A and B satisfying E=A U B there always exists a convergent sequence (x_n)-->x with (x_n) contained in one of A or B and x an element of the other.