1
vote
2answers
21 views

If $f$ is continuous on $[0, \infty)$ and uniformly continuous on $[b, \infty)$ for some $b > 0$ then $f$ is unif. continuous on $[0, \infty).

Prove that if $f$ is continuous on $[0, \infty)$ and uniformly continuous on $[b, \infty)$ for some $b > 0$ then $f$ is unif. continuous on $[0, \infty). So far I have: Let $A = [b, \infty)$ then ...
-3
votes
0answers
12 views

let E=C[X] be a normed space and T∈ L(E)… prove that.. [on hold]

Let E=C[X] be a normed space and T∈ L(E). And let $$\||P||_\ = \left\{ \sum ||P^{(n)}||_\infty, \; \; 0 \leq n \leq ∞ \right\}.$$ where $\||P||_\infty$=sup|p(x)|, 0≤x≤1 1- Justify that T:E→E ...
0
votes
2answers
36 views

Prove by induction that $(1+x)^n \geq 1+nx$ [duplicate]

Prove by induction that $\forall x \in \mathbb{R}, x \geq -1, \forall n \in \mathbb{N},n \geq 0$ that $$(1+x)^n \geq 1+nx$$ First of all I have a problem with x being a real number, how can I use ...
0
votes
2answers
61 views

Assume $f : [0, 1] \rightarrow [0, 1]$ is continuous. Show that there must be a point $x \in [0, 1]$ such that $f(x) = x$ [duplicate]

Assume $f : [0, 1] \rightarrow [0, 1]$ is continuous. Show that there must be a point $x \in [0, 1]$ such that $f(x) = x$ I am not even sure how to begin with this problem, I know that $f$ is ...
0
votes
1answer
39 views

How to prove that $\max\{f,g\}$ is Riemann integrable? [duplicate]

If f(x) and g(x) are Riemann integrable in [a,b], why $h(x)=\max\{f(x),g(x)\}$ is still Riemann integrable in [a,b]? Or maybe it is wrong?
0
votes
1answer
29 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
1
vote
2answers
33 views

Prove $f(x) = \frac{1}{x^2}$ is uniformly continuous on $[1, \infty]$

I am trying to prove this function is uniformily continuous on $[1, \infty]$, so far i have; $$|f(x) - f(x)| = |\frac{1}{x^2} - \frac{1}{y^2}| = |\frac{(x-y)(x+y)}{x^2y^2}|$$ and then, ...
1
vote
1answer
24 views

Finite subcover of pairwise disjoint open intervals

I have the following exercise: Prove that if $X$ is a countable compact subset of $ \mathbb{R}$, then for any $\varepsilon>0$ there is a finite collection of pairwise disjoint open intervals ...
0
votes
1answer
28 views

Question about a closed subspace of a complete space

Let $J$ be a closed interval. Let $C(J)$ be space of continuous functions on $J$. We know $C(J)$ is a complete metric space with metric $d(x(t),y(t)) = \max_{t \in J} |x - y | $. Consider now $$ K(J) ...
0
votes
0answers
14 views

Showing Integrability

Suppose that $I,J$ are intervals in $\mathbb{R}$ and that $F:I\to\mathbb{R}$ and $G:J\to\mathbb{R}$ are integrable. Prove that $H:I\times J\to \mathbb{R}$ defined by $H(x,y) = F(x)+G(y)$ must also be ...
0
votes
0answers
10 views

Using Implicit Function Theorem to show that F has a differentiable local inverse

Suppose that $F$ is continuously differentiable, with domain and range nonempty open set in $\mathbb{R}^n$, and that the derivative matrix of $F$ is invertible at $a$. Use the Implicit function ...
2
votes
1answer
39 views

Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$

Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$ $$ f(x) = \left\{ \begin{array}{c} -1, &0 \le x < 1 \\ 2, &1 \le x \le 2 ...
0
votes
1answer
13 views

Uniform convergence of functions and Hausdorff convergence of their graphs

Consider a sequence of continuous functions $f_n:[a,b] \to \mathbb{R}$. If their graphs $G_n$ converge to the graph $G$ of a continuous function $f$ (in the Hausdorff metric $d_H$), prove that $f_n$ ...
1
vote
0answers
13 views

Prove with Lebesgue’s Criterion for integrablility that the composition $f\circ g$ is integrable

I have this homework question regarding Lebesgue's criterion for integrability and could use a bit of help. I'm not sure if my proof is entirely correct or formal enough. Here is said question: ...
1
vote
1answer
31 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
3
votes
3answers
72 views

Evaluate $\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = \sec(u)$ substitution: $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &=& \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &=& ...
2
votes
2answers
27 views

Show that the difference quotient of $1/x^n$ exists

Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient ...
0
votes
1answer
28 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
1
vote
2answers
131 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
0
votes
1answer
38 views

Determine$\int_{a}^{b}f'(x)(f(x))^s)dx,s\in \mathbb Z$

Let $f:[a,b] \rightarrow \mathbb R$ be continuously differentable and $f(x)>0 , x\in [a,b]$. Determine the following integral: $(1)\int_{a}^{b}f'(x)(f(x))^s)dx,s\in \mathbb Z$ At ...
1
vote
1answer
22 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
0
votes
1answer
19 views

If $\mu(E_n) < \infty$ for all $n \in \mathbb{N}$ and $1_{E_n} \to f$ in $L^1$, then $f$ is a char. function of measurable set.

Problem: If $\mu(E_n) < \infty$ for all $n \in \mathbb{N}$ and $1_{E_n} \to f$ in $L^1$, then $f$ is a char. function of a measurable set. Attempt: Since $1_{E_n} \to f$ in $L^1$ , there is a ...
0
votes
1answer
37 views

Help, check the uniform continuity

(1) $f(x)=sin(1/x)$ on $(0,1]$ ? ( I know it is not uniform continuous on $(0,1)$) (2) $f(x)= xsin(1/x)$ on $(0,1]$? (3) $f(x)=sin(x^2)$ on $[0, \infty)$?
0
votes
3answers
33 views

Convergance of a sequence [closed]

Prove that the sequence $(a_n)$ converges, where$$a_n=\frac {3+n+4{n^2}}{1-n+3{n^2}}$$ for all $n\ge1$
2
votes
0answers
20 views

Question about integration and sets of measure zero

Let $Q \subseteq \mathbb{R}^n$ be a box, $f: Q \to \mathbb{R}$ be bounded, integrable on $Q$. Suppose $g: Q \to \mathbb{R}$ is another bounded function such that $f(x) = g(x)$ for any $x \in Q ...
3
votes
2answers
133 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
2
votes
2answers
64 views

Prove that integral of continuous function is continuously differentiable

Lots of things going on here. I immediately know that $F(x)$ does exist since $f$ is riemann integrable due to the fact that it is continuous. First I need to show that $F$ is continuous, then find ...
2
votes
2answers
53 views

How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
0
votes
1answer
53 views

Proof about complex exponential function forming an infinite dimensional vector space

Okay, so I sort of understand what is going on here. this has to do with the fact that the exponential function is bijective over this interval, yes? Either way, I have no idea where to start with ...
3
votes
4answers
56 views

The derivative of $(\sin(x) - 1)^{\cos(x) + 1} + (\sin(x) + 1 )^{1 - \cos(x)}$?

What is the derivative of $ f(x) = (\sin(x) - 1)^{\cos(x) + 1} + (\sin(x) + 1 )^{1 - \cos(x)}$ I don't know how to approach this one. I tried to apply the natural logarithm to both functions on the ...
1
vote
1answer
32 views

finding a complex- valued measure

1- Put $A= \Bbb D^{-}\cap {\Bbb D^{c}}^{-}$(Boundery of $\Bbb D$). Let $P= \{p|A; p=$ an analytic polynomial$\}$ and iconsider $P$ as a manifold in $C(A)$. Show that if $\mu$ is a real- valued measure ...
0
votes
4answers
42 views

Determine if the given sequence converges or diverges

Let $(x_n)$ be a sequence defined as $x_n = \frac{1}{n} \sum_{j=1}^{n} \frac{j+1}{j^2}$ . We want to know if $(x_n)$ converges. The trouble I am having here is that the sum depends on $n$. We know the ...
0
votes
0answers
33 views

Prove true or false for this problem. [closed]

Prove the following: For all sets $A,B$: $A\cup B = A$ if,and only if, $B\subset A$
0
votes
2answers
40 views

A question about using Squeeze Theorem to solve theoretical convergence question

Could you give me some hint how to deal with this question: Suppose $a_n\le b_n \le c_n$ for almost all n, $b_n\to L$, $c_n-a_n\to 0$. Prove: $a_n \to L,b_n \to L$. Well, if $a_n\to a, b_n \to b$ ...
0
votes
1answer
44 views

Inequality important in $L^p$ space

If$\,\,$ $0<p<\infty$, put$\,\,$ $\gamma_{p}=\max(1,2^{p-1})$, and show that $$|\alpha-\beta|^p \leq \gamma_{p}(|\alpha|^p + |\beta|^p)$$ for arbitrary complex numbers $\alpha$ and $\beta$. ...
2
votes
2answers
118 views

How to calculate $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$?

Could you please help me calculate this limit: $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$. My best try is : $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac ...
1
vote
1answer
41 views

positive measurable function on $[0,1]$

If $f$ is a positive measurable function on $[0,1]$, which is larger, $$\int_{0}^{1}f(x)\,\log f(x)\,dx \qquad \text{or} \qquad \int_{0}^{1}f(s)\,ds\int_{0}^{1}\log f(t)\,dt$$ Can you help me ...
1
vote
3answers
53 views

A polynomial's roots

Let $Q_n(x) = (x^2-1)^n$ and $P_n(x) = Q_n^{(n)}(x)$. Using Rolle's theorem, prove that $P_n$ has exactly $n$ roots.
2
votes
1answer
17 views

essential supremum of measurable space X

Suppose $\mu$ is a positive measure on $X$, $\quad$$\mu(X)<\infty$,$\quad$$f\in L^{\infty}(\mu)$, $\quad ||f||_{\infty}>0$, and $$\alpha_n=\int_X|f|^nd\mu \quad where \quad n=1,2,3...$$ Prove ...
0
votes
0answers
22 views

Prove that the function $f(x)=1/2^n$ is integrable on $[0,1]$

How do I prove that the function defined by $f(0)=0$ and $f(x)= 1/2^n$ if $1/2^n < x < 1/2^{n-1}$ for $n=0,1,2...$ is integrable on $[0,1]$? Using Riemann's Integrability theorom
0
votes
1answer
62 views

Application of Jensen's inequality in a proof

The problem I am on is number 7, I included 6 to show you the relevant result. I have proven number 6. Number 7, where do I start? what assumptions can I make?
2
votes
2answers
39 views

Prove: $f: (a,b) \rightarrow \mathbb{R}$ is convex iff for every triple $x_1,x,x_2 \in (a,b)$…

Assignment: Prove that a function $f : (a,b) \rightarrow \mathbb{R}$ is convex if and only if for every triple $x_1,x,x_2 \in (a,b)$ with $x_1 < x < x_2$ following inequality is satisfied: ...
2
votes
1answer
44 views

Let $f: [a,b] \rightarrow R$ be convex. If f is not constant, then the supremum of $f$ is not inside of [a,b].

I could use some help with this proof. Let $a,b \in \mathbb{R}, \ a<b \ \ f: [a,b] \rightarrow \mathbb{R}$ be convex. Prove: If f is not constant, then the point, where the supremum of ...
1
vote
1answer
39 views

How to prove : If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges.

Could you give me some hint how to prove this statement: If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges. I think, obviously wrong, that if $a_{2n}\to a$ and $a_{2n-1}\to b$ than ...
1
vote
0answers
44 views

Inductive proof about Jensen's inequality

The base case is easy. For the inductive step, i take $\lambda$ and $x$ to be as given, and then when I consider $f(\lambda_1 x_1 + . . . + \lambda_n x_n + \lambda_{n+1} x_{n+1})$ I get this is ...
3
votes
1answer
45 views

Problem with differentiable function: is it concave up when the derivative is increasing?

This makes sense to me, and I feel like it would be an easy argument IF I could use the second derivative. I'm only given that f is differentiable, NOT twice differentiable. Any help?
0
votes
1answer
6 views

CMVT and multiple derivative problem

In a previous exercise, I used the CMVT to find that for a function $f$ differentiable on an open interval $I$ containing 0 and $f(0)=0$, then $\exists c\in(0,x)$ s.t. ...
0
votes
0answers
27 views

Convergence of improper integral with $\lim_{x\to\infty}f(x)=\alpha$

I have a problem that says: Let $f:[a,\infty)\to\mathbb R$ be a function such that the improper integral $\int_a^{\infty}f(x)dx$ converges. Assume the existence of the finite limit ...
3
votes
2answers
34 views

Easy Riemann integral question

Find all functions $f$ on $[0,1]$ such that $f$ is continuous on $[0,1]$, and $\int_0^xf(t)dt$ = $\int_x^1f(t)dt$ for every $x\in (0,1)$. Attempt: Since $f$ is continuous on $[0,1]$, there exists a ...
0
votes
1answer
24 views

Question about bounded sequence with two sub-sequential limits.

Could you please give me some hint how to deal with this question. Suppose $(a_n)$ is bounded sequence with 2 sub-sequential limits. Prove : there are real numbers A and B that ...