Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

learn more… | top users | synonyms

0
votes
0answers
14 views

Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$

Suppose that $f(a)=0$. Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$ Not sure how to go about this at all. The limit definition that I am working with is $g'(a)=lim_{x \rightarrow a} ...
0
votes
0answers
28 views

Two quick questions about suprmums and real functions.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ and $\emptyset\not=X\subset \mathbb{R}$ be a bounded set. i) If $f$ is montone, does $\sup\limits_{x\in X}(f(x))=f(\sup(X)$? Is the dual statement for the ...
-6
votes
1answer
33 views

What does it mean by a set is bounded. [on hold]

Given a subset $S\subset R^m,$ what does it mean by $S$ is bounded? I missed a class so didn't get the definition... Please help.
3
votes
2answers
54 views

Proof of $(0,1)$ is not compact with usual metric.

In the proof we say $\left\{\left(\frac1n,1\right):n\geq 1\right\}$ is an infinite cover with no finite subcover. But, $(0,1)$ set also belongs to cover mentioned above. We can say $\{(0,1)\}$ is a ...
-1
votes
0answers
7 views

Some questions on convergence of measurable functions and increasing sequences.

My question relates to the proof of existence of the essential supremum, on Planetmath (http://planetmath.org/?op=getobj&id=11400&from=objects). I have some qualms with the proof so would ...
-1
votes
0answers
5 views

Boundary conditions and Lagrange Constraints in Calculus of Variations

I am trying to learn about Calculus of Variations for some time now. In many problems, there are some boundary conditions defined, for example when we want to maximize a functional ...
2
votes
2answers
33 views

Let $f\colon [a,b]\to\mathbb R$ be twice differentiable and assume $f(a)=f(b)=0$ and $f''(x)<0$ for all $x$ in $(a,b)$. Show that $f(x)>0$ on $[a,b]$.

Let $f \colon [a,b] \to \mathbb{R}$ be twice differentiable and assume $f(a)=f(b)=0$ and $f"(x)<0$ for all $x$ in $(a,b)$. Show that $f(x)>0$ on $[a,b]$. I'm not entirely sure where to ...
0
votes
2answers
23 views

Weird continuity proof

Let $I = [a,b]$ and let $f : I \to \Bbb R$ be a continuous function on $I$ such that for each $x$ in $I$ there exists $y$ in $I$ such that $| f(y)|\le | f(x)|/2$. Prove there exists a point $c$ in ...
0
votes
0answers
15 views

A dense subset in $L^2(X,\lambda)$

Suppose $S=\{f\in L^2(X,\lambda): f=\alpha_1( \chi_{A_1}-\chi_{X\setminus A_1})+\sum_{i=2}^N \alpha_ i\chi_{A_i}$ where $A_i, A_j$ are disjoint and $a_1>\max_{2\leq i\leq N} \alpha_i \}$. Is ...
0
votes
0answers
11 views

Proving Arzela-Ascoli question

I have queries on part (b): He defines $V(x,\delta) = B(x,\delta) \cap K$. And says that $E$ is dense in $K$, and $K$ is compact so $K \subset \bigcup_i^m V(x_i,\delta)$. My question is, why is ...
1
vote
1answer
16 views

Prob. 6, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to show directly that this sequence of functions does not converge uniformly?

For each $n = 1, 2, 3, \ldots$, let $f_n \colon [0,1] \to \mathbb{R}$ be defined by $$f_n(x) \colon= x^n \ \ \ \mbox{ for all } \ x \in [0,1].$$ Then $$ \lim_{n \to \infty} f_n(x) = \begin{cases} ...
1
vote
1answer
12 views

A Quick Question on Increasing Sequences and the Supremum

Let $\emptyset\not= A\subset \mathbb{R}$ and $a:=\sup A<\infty$. Now by leastness of the supremum we know that for each $n\in\mathbb{N}$ there is an $a_n\in A$ such that $a_n>a-1/n$ and so ...
1
vote
1answer
31 views

Convergence almost uniformly

Let $(X,\Sigma,\mu)$ a measure space. Let $E_n\in\Sigma$ such that $\mu(E_n)>0$ and we suppose that $f_n:=a_n\chi_{E_{n}}$, where $a_n>0$, converges almost uniformly to $0$. Then I want to prove ...
2
votes
1answer
25 views

Showing that (on $\mathbb{R}^n$) the $\|\cdot\|_\infty$ norm is weaker than any other norm

Showing that (on $\mathbb{R}^n$) the $\|\cdot\|_\infty$ norm is weaker than any other norm. I am doing past papers and the question is this: "Prove that any norm on $\mathbb{R}^n$ is weaker than the ...
3
votes
1answer
42 views

$\{\,x \in I: \phi'(x)$ $\text{does not exist}\,\}$ is at most countable.

Let $I \subseteq \mathbb{R}$ an interval. If $\phi$ is convex, then $\{\,x \in I: \phi'(x)$ $\text{does not exist}\,\}$ is at most countable. An idea please.
3
votes
0answers
19 views

Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}$ is closed and connected

Are there a closed connected subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? If we remove the condition for $C$ to be connected, we have ...
2
votes
1answer
26 views

using fundamental theorem of calculus, simple exercise

Let $f(x) = f(c) + \int_c^x h(t) \ dt$ where $f,h:[a,b] \to \mathbb{R}$ and $h$ is riemann integrable, and $c \in [a,b]$. The book I'm looking at does this: $$f'(x) = \frac{d}{dx}\left(f(c) + ...
0
votes
3answers
20 views

Find a sequence of measurable functions defined on a measurable set $E$ that converges everywhere on $E$, but not almost uniformly on $E$.

Find a sequence of measurable functions defined on a measurable set $E$ such that the sequence converges everywhere on $E$, but the sequence does not converge almost uniformly on $E$. I'm having ...
-2
votes
3answers
71 views

Find $\lim\limits_{n\rightarrow\infty}(n!)^{1/n^2}$ [on hold]

I need some help with this limit $\lim\limits_{n\rightarrow\infty}(n!)^{1/n^2}$, , I was thinking how to do, but I have no idea how to start.
1
vote
1answer
27 views

Proof of Implicit function theorem

I was trying for a simple proof for implicit function theorem on two variables.I came across a book by Dipak Chatterjee.It says as follows : $f(x,y)$ be a function of two variables and $(a,b) be a ...
0
votes
2answers
55 views

$\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$

I was trying to do this limit $$\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$$ and I cant, I will be pleased if someone give me a hint to do it.
1
vote
1answer
19 views

$\int_{\Omega\setminus A_n}f\;d\mu\to\int_\Omega f\;d\mu$ for all measurable $A_n\downarrow\emptyset$

Let $(\Omega,\mathcal{A},\mu)$ be a measure space $(A_n)_{n\in\mathbb{N}}\subseteq\mathcal{A}$ such that $A_n\downarrow\emptyset$, i.e. $A_n\supseteq A_{n+1}$ and ...
1
vote
0answers
21 views

Show that a sequence (($x_n, y_n$)) in $X \times Y$ is $e$-Cauchy if the component sequences ($x_n$) and ($y_n$) are $d_X$-Cauchy and $d_Y$ -Cauchy.

How to solve this? Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $e$ be a product metric on $X\times Y$. Show that a sequence (($x_n, y_n$)) in $X \times Y$ is $e$-Cauchy if the component ...
2
votes
1answer
18 views

Surjectivity of expanding map

Suppose that $(X, d)$ is a compact metric space and that $f: X \rightarrow X$ is a continuous function satisfying $d(x,y) \leq d(f(x), f(y))$ for all $x, y \in X$. Show that $f(X) = X$. Here is a ...
0
votes
1answer
14 views

Why is $\Delta u$ bounded, if $u\in C^2(\overline{\Omega})$ and $\Omega\subseteq\mathbb{R}^n$ is a bounded domain?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^2(\overline{\Omega})$. Why must $\Delta u$ be bounded?
1
vote
1answer
11 views

$m_{x_1x_2}\leq m_{x_2x_3}\Rightarrow f$ is convex.

Let $J \subseteq \mathbb{R}$ an interval. If $f:J \to \mathbb{R}$ such that $\forall{x_1,x_2,x_3 \in J}$ $m_{x_1x_2}\leq m_{x_2x_3}\Rightarrow f$ is convex. Where ...
4
votes
3answers
119 views

What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$?

I want to determine the limit given below: $$\lim_{n\to\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}}$$ I have tried to solve thise several times ,but with no results.I have tried using lema stolz ...
0
votes
1answer
52 views

Evaluate $\lim_{x\rightarrow \infty}(1+\frac{1}{\sqrt{x}})^{\sqrt{x}}$. Euler's Limit

Evaluate $\lim_{x\rightarrow \infty}(1+\frac{1}{\sqrt{x}})^{\sqrt{x}}$. Can I get some help? I am thinking that the limit does not exist. If you approach it from the left and then from the right, I ...
1
vote
2answers
46 views

How to show this infinite sum converges uniformly?

Let $f_k$ be a real numbers such that $\sum_{k=1}^\infty f_k < \infty$. For each $R > 0$, define the convergent sum $$v(R) = \sum_{k=1}^\infty f_k(b_k(R)e^{-ky} - c_k(R)e^{ky})$$ where $0 \leq y ...
-1
votes
0answers
33 views
0
votes
1answer
32 views

Taylor series and Maclaurin series problems

Im currently working on these two problems, and Im getting really confused with them. Can someone walk me through them? I will post the work I have so far. http://imgur.com/qXj7zC1 Here is my ...
-2
votes
1answer
21 views

bijecection mapping example and countability [on hold]

To show any bijection mapping from rational to real.
1
vote
1answer
24 views

The Essential Supremum as a Limit

Let $(X, \mathcal F, \mu)$ be a finite measure space and let $f\in L^\infty(X, \mu)$. Define $\alpha_n=\int_X |f|^n\ d\mu$. Then $$\lim_{n\to \infty}\frac{\alpha_{n+1}}{\alpha_n}=\|f\|_\infty$$ ...
0
votes
1answer
20 views

Uniform convergence when no limit function is specified.

What does it mean when someone asks you to show that a series of functions converges uniformly without specifying to what function? In the past when I've delt with uniform convergence, I've always ...
2
votes
0answers
21 views

Convex functions up to reparametrization

I would like to know if there is a standard name for functions $f:[0,1]\to\mathbb R$ with the following convexity property: $$ \forall s<t<u\qquad f(t)\leq\max\{f(s),f(u)\}$$ (the fact that ...
1
vote
3answers
54 views

If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .

Note: $x_n$ is a sequence which is not necessarily convergent. The following was my attempt. Since $\lim_{n\to \infty}a_n=a$ then $\limsup_{n\to \infty}a_n=a$ . Also ...
1
vote
2answers
24 views

If function has a given limit then to prove that function is bounded.

How to Prove that if a function $f : A \to \Bbb R$ has a limit $l \in \Bbb R$ at $c \in L(A)$, then it is bounded in a neighborhood of $c$, i.e. there exists $M \in \Bbb R$ and $\delta > 0$ such ...
0
votes
1answer
27 views

Lipschitz constant of L2 difference

What is the Lipschitz constant of $$f(A)=||Ax||_2-||Ay||_2?$$ In particular, is it $||x-y||_2$, i.e. is it true that given $A,B,x,y$, the following inequality holds: $$|f(A)-f(B)|\leq ||x-y||_2 ...
1
vote
1answer
29 views

Riemann Sum proofs

If $f$ is Riemann integrable on $[a,b]$ and $\lvert f(x) \rvert$ $\le$ $M$ for all $x$ $\epsilon [a,b]$, show that: $\lvert \int_a^b f \rvert$ $\le$ $M(b-a)$ Just started learning Riemann sums ...
1
vote
1answer
12 views

Let $f:[0,\frac{\pi}{2}]\to R$ be $f(x)=max\{x^2,cosx\}$.Prove $f(x)$ attains minimum at $x_0$ and is a sulution to $x^2=cosx$

I try to write $f(x)=\frac{1}{2}x^2+\frac{1}{2}cosx+\frac{1}{2}|x^2-cosx|$ and use the Extreme Value Theorem to show that $x_0$ exists in $[0,\frac{\pi}{2}]$, but I don't know how to show the seconde ...
1
vote
1answer
14 views

If $\lim t_n=0$, when does $\lim s_nt_n \neq 0$?

I am looking for a real-valued sequence $s_n$ such that $\lim s_nt_n \neq 0$, given that $\lim t_n=0$. Any hints?
2
votes
2answers
33 views

Derivative Definition proofs

Let $f : \Bbb R \to \Bbb R$ be defined by $$f(x)=\begin{cases}x^2, & \text{if $x$ is rational} \\ 0, & \text{if $x$ is irrational} \end{cases}$$ Show that $f$ is differentiable at $x = 0$ and ...
0
votes
1answer
35 views

A proof about boundedness for continuous functions

Let $I := [a,b]$ and let $f : I \rightarrow \mathbb{R}$ be a continuous function such that $f(x) > 0$ for each $x$ in $I$. Prove that there exists a number $a > 0$ such that $f(x) \geq a$ for ...
3
votes
1answer
28 views

A Fourier Transform Which Is Cartesian Separable

We say that the Fourier transform of a complex-valued function $f\in L^{1}(\mathbb{R}^{n})$ is separable if there exist single-variable functions $g_{1},\ldots,g_{n}$ such that $$g_{1}(\xi_{1})\cdots ...
0
votes
1answer
13 views

Boundedness Theorem for continuous functions on intervals

Just want to confirm this is a suitable proof: Assume $f$ is not bounded on $I$. So, for any $n \in \mathbb{N}$, $\lvert f(x)\rvert > n$. Since $I$ is bounded, $x_n$ is also bounded. By ...
3
votes
1answer
40 views

What can we say about the inner product of two Cauchy sequences?

Let $(x_n)$, $(y_n)$ be two Cauchy sequences in an inner a real or complex product space $X$, and let the sequence $(\alpha_n)$ be given by $$ \alpha_n \colon= \ \langle x_n, y_n \rangle \ \ \ \mbox{ ...
-4
votes
4answers
24 views

Let $f,g$ be continuous from $\mathbb R$ to $\mathbb R$ [duplicate]

Let $f, g$ be continuous from $\mathbb R$ to $\mathbb R$, and suppose that $f(r) = g(r)$ for all rational numbers $r$. Is it true that $f(x) = g(x)$ for all $x \in \mathbb R$?
2
votes
1answer
22 views

Thinking Process: a set is closed if it contains all of its limit points (--> this direction)

Is this the correct thinking process? I am thinking about a proposition that says a set is closed if it contains all of its limit points: I am just looking at --> (this direction) I suppose that ...
4
votes
2answers
196 views

Interesting counter-examples of relation between continuous and a.e. continuous function.

(a)$f$ is continuous a.e. on [0,1] (b)There exists $g$ continuous on $[0,1]$ such that $g=f$ a.e. How to prove that (a) $\nRightarrow$ (b) and (b) $\nRightarrow$ (a)? I think it can be ...
0
votes
1answer
27 views

For differentiable function where $f'(0)=a$ and $f'(1)=b$ we have that for all $c\in(a,b)$ there exists a $y$ such that $f'(y)=c$.

So what I'm trying to prove: Assume a function $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable and $f'(0)=a$ and $f'(1)=b$. Prove that for any $c\in(a,b)$ there exists a $t\in\mathbb{R}$ such that ...