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Real Analysis

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].

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?

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

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

Constructing Functions with Discontinuities

For each of the following choices of A,construct a function f:R->R that has discontinuities at every point x in A and is continuous on A^c(compliment) a-A=Z b-A={x:0<x<1} c-A={x:0<=x<=1} d-A={1/n:n belong to N}.

Real Analysis

(Composition of continuous Functions).Given f :A->R and g:B->R, assume that the range of f(A)={f(x):x belong to A} is contained in the domain of B so that the composition g o f(x)=g(f(x)) is well-defined on A.If f is continuous at c belong to A, and if g is continuous at f(c) belong to B, then g o f is continuous at c. -Supply

Determining If Series are Convergent or Divergent

(a) A certain infinite series (some of whose terms are positive and some of whose terms are negative) is known to converge, but does not converge absolutely. Explain how this is possible, by giving two such example series. (b) Determine whether each of the attached infinite series converges or diverges. See the attached file

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.

Real analysis

Definition: A set A subset or equal to R is called an F_&(F sigma) set if it can be written as the countable union of closed sets. A set B subset or equal to R is called G_&(G sigma) if it can be written asthe countable intersection of open sets. 1-Argue that a set A is a G_& (G sigma) set if and only if its complement is

Real Analysis

A set E is totally disconnected if, given any two points x,y belong to E there exist separated sets A and B with x belong to A and y belong to B and E=A U B. 1-show that Q is totally disconnected. 2-is the set of irrational numbers totally disconnected?

Real Analysis

Let A and B be subsets of R show that if there exists disjoint open sets U and V with A subset or equal of U and B subset or equal of V then A and B are separated.

Real Analysis

Show that if K is compact and F is closed then K intersection F is compact.