Tagged Questions
0
votes
1answer
42 views
Continuous function on a closed set
Let $f: F \to \mathbb R$ be defined in a closed set $F \subset \mathbb R$. Show that $f$ is continuous if and only if for all $c \in \mathbb R$, the sets $E[f \le c]=\{x \in F; f(x) \le c\}$ and $E[f ...
2
votes
0answers
45 views
Proof on showing F(x,y) is continuous by $\epsilon - \delta$ definition
The task is as follows:
Given: $$F(x,y) = \frac{xy(x^2 - y^2)}{x^2 + y^2}$$
Goal: Prove that $F(x,y)$ is continuous everywhere on the plane
Here is my attempt so far:
(1) By the ...
1
vote
0answers
42 views
How to prove $C^1$ class is a proper subset of Lipschitz class?
Let $Lip(A)$ be the set of vector-valued functions $f$ on the closed set $A\in\mathbb R^n$ such that
$$f(0)=0,$$
$$||f|| \text{ is finite, where by definition: } ||f||=\sup ...
2
votes
2answers
39 views
Holder condition for $x^\beta$
Let $f(x)=x^\beta$ (for some fixed $0<\beta<1$) be defined on $(0,1)$. It's not hard to see that $f$ is $\beta$-Holder.
How can I prove that $x^\beta$ is not $\alpha$-Holder for ...
0
votes
2answers
45 views
Continuation of smooth functions on the bounded domain
Given a bounded domain $\Omega\subset \mathbb{R}^n$ and a smooth function $f$ with bounded derivatives on $\Omega$, is it possible to extend $f$ to $\tilde{f} : \mathbb{R}^n \to \mathbb{R}$ such that ...
0
votes
1answer
23 views
Continuity of map $T: X \rightarrow X $
I have the following past exam question I've come across...
Let $(X,||\cdot||)$ be a normed space. Show that the map T:X $\rightarrow X$ given by:
$$
f_n(x) =
\begin{cases}
\dfrac{x}{||x||}
...
2
votes
0answers
44 views
Discontinuous for rationals
Show that $f\left(x\right):=\sum_{n=1}^{\infty}\frac{\left\{nx\right\}}{n^2}$, where $\left\{nx\right\}$ is the fractional part of $nx$, is discontinuous for all rationals.
I guess it would be nice ...
5
votes
3answers
80 views
Intermediate value-like theorem for $\mathbb{C}$?
Is there an intermediate value like theorem for $\mathbb{C}$? I know $\mathbb {C}$ isn't ordered, but if we have a function $f:\mathbb{C}\to\mathbb{C}$ that's continuous, what can we conclude about ...
0
votes
1answer
27 views
Proof of a special case of Banach's fixed point theorem
I have to prove the following special case of the theorem:
Let $f : I \to I$ be Lipschitz continuous on the closed (not bounded) interval $I=[0,\infty)$ with Lipschitz constant $L \lt 1$. Then $f$ ...
1
vote
1answer
96 views
Uniform limit of continuous functions bounded variation
Prove or disprove that if $f:[a,b]\rightarrow\mathbb{R}$ is the uniform limit of a sequence of continuous functions each of which is of bounded variation, then $f$ is of bounded variation on $[a,b].$
2
votes
1answer
25 views
Differentiability of first derivative of a function
If a function $f$ is differentiable on domain $D$ and $f'$ is increasing on $D$, is $f'$ necessarily continuous on $D$? Is $f'$ necessarily differentiable on $D$? Counterexamples?
From Darboux ...
0
votes
2answers
42 views
Let $f:\Bbb R^2→\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$
Let $$f:\Bbb R^2\to\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$$
i) Is $f$ continuous at $(0,0)$?
ii) Is $f$ differentiable at $(0,0)$?
I can prove that $f$ is ...
4
votes
5answers
173 views
Is there a short proof for the Intermediate Value Theorem
My final for my introductory analysis course is tomorrow and my teacher gave us a list of possible theorems to prove. If anyone could please show me a proof for The Intermediate Value Theorem that is ...
0
votes
0answers
38 views
Stieltjes integration with step function
Assume $F:[a,b]\rightarrow R$is bounded and right continuous at $a$ and $\alpha$ is the step function given by $\alpha(a)=A, \alpha(x)=B, a<x\leq b.$ Show that $f\in R(\alpha)$ on $[a,b]$ and
...
1
vote
1answer
66 views
Intermediate value property and closure of rational level sets implies continuity
Suppose $f$ satisfies the intermediate value property, i.e. if $f(a)<c<f(b)$, then there exists $a<x<b$ such that $f(x)=c$ and for every rational $r$, $S_r$ such that $f(x)=r$ is a closed ...
3
votes
1answer
58 views
Cauchy's functional equation - a generalisation? (do additive maps have to be continuous?)
If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be ...
-1
votes
2answers
49 views
What does it mean for $f$ to be continuous at $a$? [closed]
Let $a \neq 0$. Prove that $\displaystyle{f(x)=\frac{1}{x^2}}$ is continuous at $x=a$.
0
votes
2answers
52 views
How to show that $z^4$ is not uniformly continuous?
$f$ maps $z$ in $\mathbb C$ to $z^4$ in $\mathbb C$. How do I show that this function is not uniformly continuous?
2
votes
4answers
66 views
Show that $f '(x_0) =g'(x_0)$.
Assume that $f$ and $g$ are differentiable on interval $(a,b)$ and $f(x) \le g(x)$ for all $x \in (a,b)$.
There exists a point $x_0\in (a,b)$ such that $f(x_0) =g(x_0)$.
Show that $f '(x_0) ...
0
votes
3answers
47 views
Analysis Continuity
I've come across the following question and don't have any idea where to start with it.. so any help or pointers would be greatly appreciated!
Let $ C^1([0,1])$ deonte the space of continuously ...
5
votes
3answers
138 views
Continuity of $g(\theta) = \frac{1}{2\pi^2\theta^3}-\frac{\pi}{2}\cot(\pi\theta)\csc^2(\pi\theta)$ at $\theta=0$
I have the following function which I'm considering on $[0,1)$
$$g(\theta) = \frac{1}{2\pi^2\theta^3}-\frac{\pi}{2}\cot(\pi\theta)\csc^2(\pi\theta).$$
According to a graph in mathematica it is ...
2
votes
1answer
26 views
$f(x) = \inf_{y \in Y} c(x,y) - \inf_{\xi \in X} c(\xi,y) - f(\xi) \Rightarrow f$ is upper semicontinuous
Let $X, Y$ be metric spaces. Given $c: X \times Y \mapsto \mathbb{R}$ continuous, define
$$ f(x) = \inf_{y \in Y} \left( c(x,y) - \inf_{\xi \in X} (c(\xi,y) - f(\xi)) \right).$$
Then is $f$ upper ...
0
votes
2answers
54 views
Is level set of sum of two continuous functions a closed set?
$f^i: R^{n}\to R^{n}$ is a continuous function for $i=1,2$.
Let
$$M=\{(x,y)\in R^{2n}~|~f^1(x)+f^2(y)=0\}$$
Is $M$ a closed set? If not, can you give a counter example.
1
vote
1answer
72 views
Showing a function is discontinuous
I've used matlab to get some idea of how the following function behaves:
$$g(\theta) = \frac{2}{\theta^3} - \frac{\pi\cos(\pi\theta)}{2\sin^3(\pi\theta)}.$$
It appears that it is discontinuous at ...
3
votes
0answers
74 views
Prove that if $f$ is uniformly continuous then the one sided limit $\lim_{x\to 0^+} f(x)$ exists. [duplicate]
If $f(x)$ is a continuous function on $(0,1]$, prove that if $f$ is uniformly continuous, then the one sided limit $\lim_{x\to 0^+} f(x)$ exists.
5
votes
2answers
81 views
A question on a Lipschitz function
This is the problem:
Prove or disprove the following statement:
If $f:[0,+\infty]\rightarrow\mathbb{R^+}$ is a Lipschitz function and not bounded, then it has necessarily $\lim_{x\to+\infty} f(x) = ...
2
votes
3answers
122 views
Uniform Continuity of $f(x)=x^3$
1.)Determine whether $f(x)=x^3$ is uniformly continuous on [0,2)
So far, I have $\delta$ = 2 and $\epsilon$ = 8, and plan on using the sandwich theorem with $x^2$ and eventually equating $\delta = ...
1
vote
1answer
32 views
Equintinuity of bounded linear functions equivalent to uniform boundedness
The claim is the following:
Every family of bounded linear functions is equicontinuous if and only
it is uniformly bounded.
Equicontinuity is defined here. Any suggestions about this?
I only ...
1
vote
1answer
52 views
Regarding Hölder continuity
Let $\alpha \geq 0$. We say that $f \colon D \to \mathbb{R}^m$ is $\alpha$-Hölder continuous if there is a constant $c$ such that for each $x,x_0\in D$, $|f(x) - f(x_0)| \leqslant c\cdot |x - ...
2
votes
2answers
118 views
Prove that $f$ is discontinuous at $(0,0)$
Let $f$ be defined by
$$ f(x,y) =
\begin{cases}
\biggl\lvert \frac{y}{x^2} \biggr\rvert e^{-\bigl\lvert \frac{y}{x^2} \bigr\rvert} , \quad \text{ if $x \neq 0$} \\
0, \qquad \qquad \quad \text{if $x ...
0
votes
0answers
88 views
Prove that $f$ is continuous if and only if $f(\cdot+t) \to f$ pointwise as $t \to 0$
Let $f:\mathbb{R}\to \mathbb{C}$ be a function.
Prove that $f$ is uniformly continuous if and only if $f(•+t) \to f$ in $L_{∞}$ as $t\to0$
Prove that $f$ is continuous if and only if $f(•+t) \to f$ ...
7
votes
1answer
117 views
$f:[a,b]\to(a,b)$ be continuous how prove $f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)(c+\frac{nd}{2})$
let $f:[a,b]\to(a,b)$ be continuous how prove $\forall n\in\mathbb N$ $\exists d\gt0$ ,$\exists c\in(a,b) $ such that $$f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)\left(c+\frac{nd}{2}\right)$$thanks in advance ...
0
votes
1answer
52 views
I don't understand why the contrapositive part of the proof of continuity holds
So, assuming whoever can answer this knows the first part of this proof, that is, showing that for all$ \epsilon\gt0$ and for all $ p\in M$ there is a $\delta\gt0$. Also there is an $ x\in M$ such ...
3
votes
1answer
96 views
Proving continuity of an integral
I have the following function:
$$I_n(a)=\int_{-\infty}^{\infty}x^6e^{-x^2}\operatorname{sech}^n(ax)dx$$
where $\operatorname{sech}(x)=\frac{2}{e^x+e^{-x}}$ is the hyperbolic secant.
Clearly, the ...
2
votes
1answer
171 views
Proof that a continuous function is bounded below
I have this question:
Assuming the theorem that a continuous real-valued function on a
closed bounded interval is bounded and attains its bounds, prove that
if $f\colon\mathbb R\to\mathbb R$ ...
3
votes
0answers
31 views
Differentiable function which is nowhere continuously differentiable [duplicate]
Possible Duplicate:
How discontinuous can a derivative be?
$x^2\cos(1/x)$ is the standard example for a differentiable function whose derivative is not continuous at $x=0$.
But is there ...
4
votes
0answers
77 views
Function that is discontinuous only for integer fractions
I have this question:
Find a function $f :\mathbb R \to\mathbb R$ which is discontinuous at the points of the
set $\{\frac1n : n \text{ a positive integer}\} \cup \{0\}$ but is continuous ...
0
votes
0answers
37 views
What are the features of $n/d, n \rightarrow d, d \rightarrow \infty; n, d \in \mathbb{N} $?
What are the features of $n/d, n \rightarrow d, d \rightarrow \infty; n, d \in \mathbb{N} $?
What is the value of $\lim_{n \rightarrow d, d\rightarrow \infty} (n/d)$? What is the function's range? ...
2
votes
1answer
54 views
Continuity of $x+y$ and $xy$ in $\mathbb{R}^{\infty}$
How can I show (or where can I find) that in $\mathbb{R}^\infty$: $f(\textbf{x},\textbf{y})=\textbf{x}+\textbf{y}$, $g(\textbf{x}, k)=k\cdot \textbf{x}$ are continuous functions?
($g$ is from ...
6
votes
1answer
135 views
Is the following function continuous at $x = 0$?
Define $f: \mathbb{R} \to \mathbb{R}$ by
$$ f(x) = \cases{
x - 1 \ \ \text{ if } x \in \mathbb{Q}
\\1 - x \ \ \text{ if } x \not\in \mathbb{Q}.
}$$
I'm trying to prove whether or not $f$ is ...
2
votes
2answers
102 views
Continuity of an Analytic Function
I am trying to show that the analytic function, $f: (0, \infty) \to \mathbb{R}$, defined by
$ f(x) = \sum \limits_{n = 1}^\infty ne^{-nx} $
is continuous.
I don't have much experience with ...
1
vote
1answer
54 views
Continuity of integer part function
Check the continuity of $f:[0,1]\to \Bbb R$:
$f(x) =
\cases{
0 & \text{if } x=0\cr
\dfrac{1}{\left[\frac{1}{x}\right]} & \text{if } 0<x\le 1
}$
where $\left[\dfrac{1}{x}\right]$ is the ...
1
vote
2answers
55 views
Question on continuity of functions from $X\times X\rightarrow Y$
I am stuck on the following problem, which I do not believe to be so difficult.
Let $X$ and $Y$ be Banach spaces. Let $f:X\times X\rightarrow Y$ be a function such that for any fixed $x_0$, ...
1
vote
0answers
72 views
Let $y:[0,1] \rightarrow \mathbb R$ be a twice continuously differentiable function such that $y''(x)-y(x)<0 $
I came across the following problem that says:
Let $y:[0,1] \rightarrow \mathbb R$ be a twice continuously differentiable function such that $y''(x)-y(x)<0 $ for all $x \in (0,1)$ and ...
3
votes
3answers
130 views
Why Norms are Continuous with details
Please one person describe why norms are continuous function by mathematical symbols.
0
votes
1answer
54 views
Consider the three subsets of $\mathbb R^2$
I have been trying to solve the problem. Since the function is defined for the values taken from the three given sets, it seems that $(2)$ is one of the right options. But, I cannot determine the ...
2
votes
1answer
54 views
Suppose the function $f:\mathbb R \rightarrow \mathbb R$ has left and right derivatives at $0$.
I have been trying to solve the following problem:
Suppose the function $f:\mathbb R \rightarrow \mathbb R$ has left and right derivatives at $0$.Then at $x=0$, which of the following options is ...
0
votes
2answers
411 views
Prove or disprove that $f(x) = \sin x/x$ is uniformly continuous over the interval $(0,1)$?
I am thinking of $\lim_{x \to 0} \ {f(x)} = 1$ but I am confused as $x=0$ is not in the domain and also I want to write an $\epsilon$-$\delta$ proof. So any help is much appreciated!
Thanks!
0
votes
1answer
57 views
Is this function Measurable?
Suppose $f$ is a continuous function defined on $[0,1]$. Let $\operatorname{sgn}(f)$ denote the signal function of $f$. Is $\operatorname{sgn}(f)$ a measurable function?
0
votes
1answer
97 views
Let $f$ be a real valued sequentially continuous function relative to a closed bounded interval $I=[a,b]$. Prove that the set $f(I)$ is bounded above
The hint that I've been given is: for each n in the naturals, use the assumption that $n$ is not an upper bound for $f(I)$ to choose a sequence of $x_n$ (from $n=1$ to infinity) in $I$; then apply ...
