0
votes
0answers
33 views

Requirements for integration by parts/ Divergence theorem

In order to use the integration by parts formula(or more generally the divergence theorem) for functions of several variables $$\int_{\Omega} \nabla u\cdot v d \Omega = \int_{\partial \Omega}(u(v ...
1
vote
2answers
49 views

Proving $ f(x)=(\frac {\sin x} {x})^{\frac {1} {x^2} }$ is uniformly continuous on $(0,1]$

Prove that $f(x)=\Large(\frac {\sin x} {x})^{\frac {1} {x^2} }$ is uniformly continuous on $(0,1]$. Basically what I need to show here is that there is a limit 'from the right' for $x=0$ so the ...
2
votes
1answer
33 views

$\varepsilon$-$\delta$ proof of continuity of floor function $\lfloor x\rfloor$

I would just like to ask someone to confirm or correct the following 'proof' of continuity of the floor function. Let $\varepsilon>0$ be given. Set $\delta:=\min\lbrace x-\lfloor x\rfloor,\lceil ...
4
votes
1answer
71 views

Proof verification: $\int_a^x f(t) \text{dt}=0$, $f$ is continuous at $x$. Prove that $f(x)=0$

Let $f:[a,b]\to R$ be an integrable function such that for all $x \in[a,b]$, we have $\int_a^x f(t) \text{dt}=0$. Show that if $f$ is continuous at $x \in [a,b]$, then $f(x)=0$. My attempt: argue ...
3
votes
1answer
25 views

Show for whih values this following function is continuous

For the function $f: [0,2 \pi] \rightarrow \mathbb{R}$ ,state at which points $c \in [0, \pi]$ is $f$ continuous or discontinuous. $$f(x)=\begin{array}{cc} ( & \begin{array}{cc} ...
0
votes
1answer
32 views

Continuity proof of two-variable function.

The Assignment Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} ...
1
vote
3answers
28 views

Where does my proof of uniform continuity fail?

I am trying to prove that $f:R \to R f(x)=\sin x$ is uniformly continuous. I have said: Fix $\epsilon > 0$ and $\delta=\epsilon$ $|\sin x - \sin y| \le |\sin x| - |\sin y| \le 1 - 1 = 0 ...
1
vote
1answer
38 views

Uniform continuity of $\arctan x$

Check if $\arctan x$ is uniformly continuous on $\mathbb R$ If I'll show that it's contious on $[0,\pi/2]$ then because it's periodic it would be continuous on $\mathbb R$. So by the definition ...
4
votes
1answer
29 views

Prove there exists a point $c$ such thst $f(c)=c$ for the following function

If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function with $f(0)=2$ and $|f'(x)| \leq 1/2$ for all $x$ then there is a point $c$ such that $f(c)=c$ . My Attempt Let ...
0
votes
2answers
61 views

If $f,g$ are uniformly continuous prove $f+g,fg$ are uniformly continuous

Suppose $f:E \rightarrow \mathbb{R}$ and $g:E \rightarrow \mathbb{R}$ are uniformly continuous, where $E$ is a subset of $\mathbb{R}$. Show that $f+g \ \ and \ \ fg$ are uniformly contiuous, what ...
0
votes
1answer
17 views

Using Properties of a Dense Set to prove characteristics of a continuous function

1. If $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous and $f(x)=0$ for all $x$ in a dense set $E$, then $f(x)=0$ for all $x \in \mathbb{R}$ 2. If $f:\mathbb{R} \rightarrow \mathbb{R}$ and ...
0
votes
1answer
17 views

For what points $c$ in $\mathbb{R}$ is $f$ continuous?

Let $X \subset \mathbb{R} $ be a fintie set and define $f:\mathbb{R} \rightarrow \mathbb{R}$ by $f(x)=1$ is $x \in X$ and $f(x)=0$ otherwise. At which points $c \in \mathbb{R}$ is $f$ continuous? ...
1
vote
1answer
59 views

Proofs about continuity and convergence in topological spaces

I'm working on the following exercise: Let $f:(X,T)\to(Y,S)$ and $x\in X$. Prove that if $f$ is continuous at $x$ then if a sequence $\{x_n\}$ converge to $x$ we have $f(\{x_n\})\to f(x)$, show ...
1
vote
0answers
25 views

Check my answer - show a function is integrable and find the integral

Let $Q =[0,1]$x$[0,1]$. Let $f: Q \to \mathbb R$ defined as such: if $(x,y) \in \mathbb Q$x$\mathbb Q$ then $f(x,y)=\frac{1}{n_1}+\frac{1}{n_2}$ where $x=\frac{m_1}{n_1}$ and $y=\frac{m_2}{n_2}$ are ...
0
votes
2answers
36 views

If f is continuos on an interval, is it then uniformly continuous

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I now know that it is not. Can someone give me a proof ...
0
votes
1answer
26 views

If $f$ is continuous on $(0,5)$, is it uniformly continuous on same interval

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I believe it is. I now know that it is not. Can someone ...
1
vote
2answers
57 views

$\varepsilon$-$\delta$-definition for continuity of $x^n$

Show that $f:\Bbb R\to\Bbb R,x\mapsto x^n$ with $n\in\Bbb N$ is continuous in $x_0=0$ using the $\varepsilon$-$\delta$-definition. We assume that $$\forall ...
0
votes
0answers
53 views

$f \in \mathcal{R}(\alpha)$ on $[a,b]$, then $\exists P_n$ s.t. $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$.

Assume $f \in \mathcal{R}(\alpha)$ on $[a,b]$, and prove that there are polynomial $P_n$ such that $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$. This is what I have, ...
3
votes
1answer
149 views

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...
0
votes
0answers
177 views

$\{f_n\}$ equicontinuous sequence of functions on compact $K$, converges pointwise on $K$ then converges uniformly on $K$.

Suppose $\{f_n\}$ is an equicontinuous sequence of functions on a compact set $K$ and $\{f_n\}_{n=1}^{\infty}$ converges pointwise on $K$. Prove that $\{f_n\}_{n=1}^{\infty}$ converges uniformly on ...
1
vote
1answer
42 views

Lower semi-continuity of a convex functional on $L^1(\Omega,[0,1])$

Let $\Omega$ be a bounded domain and $f:\Omega\times[0,1]\to[0,\infty]$ be such that $x\mapsto f(x,u)$ is measurable for every $u$, $u\mapsto f(x,u)$ is continuous and convex for a.e. $x$. Furthermore ...
0
votes
1answer
62 views

Prove $mx+b$ is continuous at any point in $\mathbb{R}$

I need to prove, $mx+b$ is continuous at any point in $\mathbb{R}$ Now, as I have thought of there's 2 possible cases: 1) $m = 0$ 2) $m \neq 0$ So for case #2, $m < 0 \vee m > 0$ , and we ...
0
votes
0answers
119 views

Verification of $\epsilon$-$\delta_\epsilon$ definition of continuity proof

Show that the function $$ f(x) = \begin{cases} \frac{1}{q}, & \text{if $x = \pm \frac{p}{q}$ in the lowest terms with $p, q \in N$} \\ 0, & \text{if $x \in R - Q$} \end{cases}$$ ...
1
vote
1answer
48 views

Introduction to Analysis: Continuity and Sequences

The following is a Theorem the instructor gave up. Let $f(x)$ be defined for $x \approx a$, and suppose that for all {$x_n$} such that $x_n \rightarrow a, x_n \neq a$, we have $\lim_{x \rightarrow ...
1
vote
1answer
35 views

Introduction to Analysis: Bisections

I feel as though I may be over thinking this problem, at the same time I feel like I may be under thinking it. Use a bisection argument to prove that if $f:[a,b] \rightarrow \mathcal{R}$ is ...
2
votes
1answer
137 views

How to show that time-dependent norm is continuous (please verify my proof)

For each $t \in [0,T]$, let $H_t$ be a Hilbert space. Suppose for each $t$, the operator $T_t:H_0 \to H_t$ is a linear homeomorphism with inverse $T_{-t}:H_t \to H_0$ also linear homeomorphism. ...
1
vote
1answer
74 views

Introduction to Analysis: Multiplicatively Periodic

I was given this problem. Been stuck on it for a while but I have an idea. The problem reads: Call a function "multiplicatively periodic" if there is a positive number $c \neq 1$ such that $f(cx) ...
1
vote
1answer
44 views

Introduction to Analysis: Locally and Actually Constant

I was given this problem for homework. I more a less understand it. I just need to somehow finalize my ideas. The problem reads: Prove that a function which is locally constant on $[0,1)$ is ...
0
votes
1answer
108 views

Introduction to Analysis: Continuity and Limits

My coworker and I were looking at a problem for our Real Analysis class. It reads: Call a function "multiplicatively periodic" if there is a positive number c $\neq$ 1 such that $f(cx) = f(x)$ for ...
2
votes
2answers
225 views

Is this epsilon-delta proof correct?

Consider the function $f:\mathbb{R}\rightarrow\mathbb{R}$ $$f(x)=\begin{cases}x,\ x\in\mathbb{Q} \\ -x,\ x \notin \mathbb{Q}.\end{cases}$$ I'm trying to prove that for all $a \neq 0$, $\lim_{x \to ...
0
votes
1answer
383 views

Show that this real function is Lipschitz continuous

I have this excersize: Let $I$ be an interval in $\Bbb R$ and $f:I\to \Bbb R$ a differentiable function such that $sup_{x\in I}|f'(x)|<\infty$. Show that $f$ is Lipschitz continuous. Well, I ...
0
votes
1answer
122 views

Proof that $\max(x_1,x_2)$ is continuous

I haven't done a proof in years and I've become a little stuck on this, I'd appreciate it if somebody could tell me if I've approached the problem correctly... Question: Prove that the following ...
3
votes
1answer
118 views

$O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb R).$

Let $M(n,\mathbb R)$ be endowed with the norm $(a_{ij})_{n\times n}\mapsto\sqrt{\sum_{i,j}|a_{ij}|^2}.$ Then the set $O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb ...
1
vote
1answer
65 views

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous.

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous. ($M(n,\mathbb R)$ is identified with $\mathbb R^{n^2}$ as a normed liner space.) ...