### 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.

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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.

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].

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

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

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

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).

A function is increasing on A if f(x)<=f(y) for all x <y in A. Show that the intermediate value theorem does have a converse if we assume f is increasing on [a,b].

See attached... Let f(x) be a function defined for x>=1, f(1)=1 and df/dx=1/(x^2+f(x)^2) Prove that limf(x) exists and is less than 1+(pi/4)

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].

Decide whether or not the following infinite series converge, in each case prove your result (by using theorems) *(Please see attachment for series)

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

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?

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

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.

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

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

Please see the attached file for the fully formatted problem. Find the sum in the closed form: 1/2! + 2/3! +...+ n/(n+1)!

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 .

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.

Show that if x=lim a_n for some sequence (a_n) contained in A satisfying a_n not = x,then x is a limit point of A.

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.

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.

Give an example to show that its possible for both Sum of x_n(sum sign) and sum of y_n to diverge but for Sum of x_n y_n to converge.

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

If {G1,G2,G3,...} is a countable collection of dense, open sets then the intersection (U top infinity bottom n=1)G_n is not empty.

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

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?

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.

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

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