0
votes
1answer
17 views

Continuity along different spaces

1) Say I have a function that is continuous along $\mathbb{R}.$ Would that function be then continuous along $\mathbb{Q}$ ? How about the other way around? 2) If I have two functions that are not ...
1
vote
3answers
91 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
0
votes
0answers
25 views

Proving $\cos$ is Lipschitz continuous with $L=\frac{\sqrt3}2$ on $[-\frac12,1]$, using $\frac{\sqrt3}2=\cos\frac\pi6=\sin\frac\pi3$

I'm working my way through some analysis exercises to gain a better understanding and I stumbled upon an exercise where I could really use a hint. The task is to show that the inequality $|\cos ...
0
votes
0answers
9 views

Upper semicontinuous function and equivalent statements

Problem Let $f:\mathbb R^n \to \overline{\mathbb R}$, then the following statements are equivalent: (1) $f$ is upper semicontinuous; (2) for every $t \in \overline{\mathbb R}$, $\{x \in \mathbb ...
2
votes
1answer
35 views

Does differentiability imply absolute continuity? [duplicate]

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is a function which is (i) differentiable at all $x \in (a,b)$ (ii) the right-derivative at $x=a$ exists and the left-derivative at $x=b$ exists. Does it ...
3
votes
1answer
47 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
7
votes
2answers
140 views

$f:\mathbb{R}\to \mathbb{R}$ continuous and $\lim_{h \to 0^{+}} \frac{f(x+2h)-f(x+h)}{h}=0$ $\implies f=$ constant.

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function with the property that $$\lim_{h \to 0^{+}} \dfrac{f(x+2h)-f(x+h)}{h}=0$$ for all $x \in \mathbb{R}$. Prove that $f$ is constant.
0
votes
1answer
19 views

Continuity of function does not imply continuity of extension

Let $f$ be increasing on a dense subset $D$ of $\mathbb{R}$, and define $\tilde{f}$ on $x\in\mathbb{R}$ $\tilde{f}(x):=\inf_{x<t\in D}f(t)$. Show that the continuity of $f$ on $D$ does not imply ...
3
votes
0answers
21 views

Existence of increasing, smooth modulus of continuity

First, recall the definition: Given a function $f:M\to N$, where $M$ and $N$ are metric spaces, a modulus of continuity for $f$ is a function $\omega:[0,\infty)\to[0,\infty)$ such that ...
1
vote
1answer
42 views

Which function is not uniformly continuous? [closed]

Which of the following functions is not uniformly continuous? $$A.\ \ \ \frac{1}{x}, \ \ \ x \in [1, +\infty)$$ $$B. \ \ \ \ \ \ \ \frac{1}{x}, \ \ \ x \in (1,2)$$ $$C. \ \ \ \ \ \ \ \ \frac{1}{x}, ...
1
vote
0answers
13 views

Continuity of the solution to a matrix PDE (mapping of a parameter to solution)

I'm considering the following PDE in $\Phi$: $\frac{\partial \Phi(t,s)}{\partial t}$ + $sR\frac{\partial \Phi(t,s)}{\partial s}$ + $\frac{1}{2} s^2 M \frac{\partial^2 \Phi(t,s)}{\partial s^2}$ + ...
0
votes
1answer
67 views

The linearity of $D \beta : \mathbb{E_1} \times \mathbb{E_2} \rightarrow \mathcal{L}(\mathbb{E_1} \times \mathbb{E_2},F)$

Let $\mathbb{E_1}, \mathbb{E_2}$ and $\mathbb{F}$ normed spaces of finite dimensions and $\beta : \mathbb{E_1} \times \mathbb{E_2} \rightarrow \mathbb{F}$ is one bilinear function. Then $D \beta : ...
4
votes
0answers
155 views

Differentiability-Related Condition that Implies Continuity

I previously asked a related question here that I did not phrase as I intended. This is a revision of that question: It is a well-known fact that differentiability implies continuity. And, for ...
0
votes
2answers
39 views

Question about limit and continuity

I have that $u_0>0$ , $u_n=u_n^+-u_n^{\raise{1pt}{-}}$ and $u\mapsto u^{±}$ is continuous if $u_n\rightarrow u_0$ why we have that $u_n^+\rightarrow u_0$ and $u_n^{\raise{1pt}{-}}\rightarrow 0 $ ...
0
votes
1answer
12 views

continuity of a function f = (f_1,f_2) in a product topology if f_1 and f_2 are continous

Say $X$, $Y_1$ and $Y_2$ are topological spaces. Let $f_1 \; X \to Y_1$ and $f_2 \; X \to Y_2$. If $f\; X \to Y_1 \times Y_2 $ $f(x) = (f_1(x), f_2(x))$ $Y_1 \times Y_2$ is a topological space with ...
0
votes
2answers
42 views

showing that $f(x,y)$ is continuous at $(0,0)$

Let $$f(x,y) = \begin{cases} 0, & \text{if $y \le 0$, $y \ge x^2$ } \\[2ex] 1, & \text{if $0 \lt y \lt x^2$ } \\ \end{cases}$$ Show that $f(x,y) \to 0$ as $(x,y) \to (0,0)$ along any ...
0
votes
1answer
44 views

Construct a continuous function which has no derivative almost everywhere.

Georg Cantor is famous for the first set theory (in "naive" terms) and the diagonal argument. However Cantor is also credited with the Cantor Set and for constructing a continuous function which has ...
1
vote
0answers
13 views

Finding the continuity of the mapping of a solution to a PDE to its partial derivative

Here is a modified version of the Black-Scholes PDE: $\frac{\partial \phi(t,S,i)}{\partial t}$ + $r_iS\frac{\partial \phi(t,S,i)}{\partial S}$ + $\frac{1}{2} \sigma^2_i S^2 \frac{\partial^2 ...
1
vote
1answer
44 views

$V$ is open , then $V=\{x\in \mathbb R:f(x)>0\}$ for some continuous function $f$

Let $V$ be a non-empty open set of real numbers , then how to prove that there is a continuous function $f:\mathbb R\to \mathbb R$ such that $V=\{x\in \mathbb R:f(x)>0\}$
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 ...
0
votes
1answer
31 views

Proving a function is not uniformly continuous.

I am using the definition: $(∃ε > 0)(∀n ∈ N)(∃ x_n, y_n ∈ (0,1])[(|x_n − y_n| < δ_n =1/n) ∧ (|f(x_n) − f(y_n)| ≥ ε)]$ to prove that $1/x^2$ is not uniformly continuous. In the solution I am ...
1
vote
0answers
24 views

Does right continuity imply only countably many discontinuities? [duplicate]

Does right continuity imply only countably many discontinuities? That is, if $f:\mathbb{R}\rightarrow \mathbb{R}$ is right continuous then does it only have countable many discontinuities? Thanks
1
vote
1answer
38 views

Question about limits and Mean Value Theorem

Let $f:(a,b) \rightarrow \mathbb{R}$ and $g:(a,b) \rightarrow \mathbb{R}$ be differentiable on (a,b) with $g'(x) \neq 0$ for all $x$ in $(a,b)$. Suppose $\lim_{x \to b-}\dfrac{f'(x)}{g'(x)}$ ...
1
vote
0answers
42 views

Is function continuous at 0?

Is $f:[0, \infty) \rightarrow \mathbb{R}$ $f(x)=[x^{1/2}]$ continuous at 0? My Attempt Now using the limit method, and as the function is only defined on $[0, \infty)$ $$\lim_{x \to +0}=\lim_{x ...
0
votes
1answer
12 views

About maximum function and continuity

Let $\bar{x}\in\mathbb{R}^n$, $R>0$, and $P$ metric space. If $f:\bar{B}(\bar{x},R)\times P\rightarrow\mathbb{R}$ is a continuous function. We define $F:P\rightarrow\mathbb{R}$ by ...
1
vote
1answer
49 views

Show that $f*(x) = \sup \{ f(y) : a \leq y \leq x \}$ is a non-decreasing continuous function

I am currently working on a problem and stuck on it. Here is the problem (it comes form Elementary analysis, the theory of Calculus by K. Ross P.153): Q: Let $f$ be a continuous function on [a,b]. ...
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 ...
2
votes
0answers
22 views

A basic question about upper hemicontinuity

Given a correspondence $f:X\rightarrow 2^X$, suppose X is a closed simplex in $\mathbb{R}^n$, and $f$ is compact-valued. We say $f$ is upper hemicontinuous if, $\forall x\in X$ and every open subset ...
0
votes
0answers
36 views

Example of continuous function without fixed point.

I need to find an example of a continuous function without a fixed point, and this is what I've come up with: As {1} is not in the (co)domain, I can evade all $x$ for which $f(x)=x$ up until I ...
1
vote
2answers
36 views

Prove the following statements

Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous with $f(0)=0$ and $f(1)=1$. For the following you may apply standard results without proof provided you state them carefully; $(1)$ If ...
0
votes
1answer
24 views

If $g$ is continuous and $f$ is s.t $f=g$ for $|x|<1$ then $f$ is continuous at 0

If $g:\mathbb{R} \rightarrow \mathbb{R}$ and $f:\mathbb{R} \rightarrow \mathbb{R}$ is such that $f(x)=g(x)$ for all $|x|<1$ then $f$ is continuous at 0. Attempt; My claim is this statement ...
0
votes
2answers
66 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
37 views

Give an example of a continuous function with this property

Find an example of a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ with the following property, For every $\epsilon >0$ there exists $\delta > 0$ such that $|f(x)-f(y)|< ...
1
vote
0answers
23 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
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
34 views

If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself)

$(i)$ Given a function $f:E \rightarrow \mathbb{R}$ define $|f|:E \rightarrow \mathbb{R}$ by $|f|(x)=|f(x)|$ for $x \in E$ show if $f$ is continuous at $c \in E$ then so is $|f|$. $(ii)$ Now ...
0
votes
2answers
32 views

At what points is this piecewise function continuous?

At which $c \in \mathbb{R}$ is the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined be $$f(x) = \begin{cases} x & x \in \mathbb{Q}\\ 1-x & x \notin \mathbb{Q} ...
2
votes
1answer
23 views

If $f$ is not continuous , there exists $x_n$ where $x_n \rightarrow n$ but $f(x_n)$ doesnt go to $f(c)$

If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a function that is not continuus at $c$, show that there exists a sequence $(x_n)$ such that $\lim_{n \to \infty}x_n=c$, but such that $f(x_n)$ does not ...
1
vote
1answer
24 views

Basic proof of continuity from definition (With Answer to my own question)

Prove directly from definition that $$f(x)=3x+25$$ is continuous everywhere. NOTE: I will be posting a solution to this problem below.
0
votes
2answers
51 views

Definition of continuity question

Hey sorry about the picture, its of an example in a lecture slide. Just a quick question is I understand how they get the answer (in red) if they are told the constraint that delta must be less ...
1
vote
2answers
33 views

If $g$ is discontinuous and $fg$ is continuous then $f$ is continuous

Prove or Provide Counterexample; Suppose $f:(a,b)\rightarrow \mathbb{R}$ is such that for all discontinuous functions $g:(a,b)\rightarrow \mathbb{R}$ the product $fg$ is continuous.Then $f$ is ...
1
vote
1answer
50 views

Cases when f(x)g(x) is continuous

Prove or produce a counterexample for the following; $(i)$ Suppose $f:(a,b) \rightarrow \mathbb{R}$ is such that whenever $g:(a,b) \rightarrow \mathbb{R}$ is continuous then so is the product ...
0
votes
0answers
26 views

Discontinuity of floor function

I am still getting confused with showing discontinuity of functions, here is my attempt at a question , if someone could check to see if I am going about it in the correct way. $f: \mathbb{R} ...
1
vote
0answers
19 views

$f\mapsto \sum_{n\in \mathbb Z} |\widehat{F(f)}(n)|$ lower semi continuous?

Let $T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
1
vote
2answers
37 views

How to prove that this function is continuous?

If $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous on the rectangle $R=[a,b] \times [c,d]$, prove that the function $g(x) := \int\limits_{c}^{d} f(x,y) dy$ is continuous on $[a,b]$. Thanks in ...
2
votes
1answer
46 views

Continuity Problem, prove that $f(r)=r^{2013}$ [closed]

Let $f$ : $R$ → $R$ be differentiable such that for all $x \in R$, $$f(1 − f(x)) = 1 − x^9.$$ If $f(1) = 0$ and $f′(1) < 0$, then prove that there exists $r \in R$ such that $f(r) = r^{2013}$. ...
0
votes
1answer
45 views

Is the function continuous?

If $f_j:\Bbb R\to\Bbb R$ is continuous for $j=1,\ldots,n$, then so is the function $g$ defined by $$g(x)=\max\{f_1(x),\ldots,f_n(x)\}.$$ If $f_j:\Bbb R\to\Bbb R$ is continuous for each $j\in\Bbb ...
0
votes
2answers
42 views

Prove the statements or show counterexample

If $f\colon\mathbb R\to\mathbb R$ is everywhere differentiable and $\lim_{x\to\infty} f(x)=\lim_{x\to-\infty}f(x)=\infty$ then there is a point $x_0$ where $f'(x_0)=0$. and If $f\colon\mathbb ...
0
votes
1answer
19 views

Proving a function is bounded above.

Hi all, while doing this question ,I feel that I understand the concept of the question, but can't seem to formulate it into a viable answer. If the limit as $x \rightarrow \infty$ is the same as $x ...