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.
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.
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.
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
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].
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.
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
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.
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].
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
Let g be defined on all of R.if A is a subset of R define the set g^-1 (A) by g^-1 (A)={x belong to R:g(x) belong to A}. Show that g is continous if and only if g^-1 (O) is open whenever O subset or equal to R is an open set.
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).
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).
Prove that if A function f:A->R fails to be uniformly continous on A if and only if there exists a particular e>0(epsilon) and two sequences (x_n) and (y_n) in A satisfying absolute value of x_n - y_n -->0 but absolute value of f(x_n)-f(y_n)>=e
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}.
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
(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.
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.
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}.