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Real Analysis

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.

Real Analysis : Subsets and Limits

Let f and g be functions defined on a domain A subset or equal to R, and assume lim_x-->c f(x)=L and lim_x-->c g(x)=M for some limit point c of A then, 1-lim_x-->c k f(x)=kL for all k belong to R. 2-lim_x--> [f(x)+g(x)]=L+M 3-lim_x-->c [f(x)g(x)]=LM 4-lim_x-->c f(x)/g(x)=L/M provided M not = 0

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.

Real Analysis

Show that if K is compact, then sup K and inf K both exist and are elements of K

Real analysis

Let A be bounded above so that s= sup A exists show that s belong to closure A(A over it bar)

Real analysis

Let x belong to O, where O is an open set.If (x_n) is a sequence converging to x prove that all but a finite number of the terms of (x_n) must be contained in O.

A Limit

Find the following: the limit as x approaches 0 of (tan 3x * cot 2x)

Real Analysis

A set F subset or equal to R is closed if and only if every Cauchy sequence contained in F has a limit that is also an element of F.