2
votes
1answer
28 views

A continuous function that attains neither its minimum nor its maximum at any open interval is monotone

Let $f: \mathbb R\to \mathbb R$ be a continuous function such that $f$ attains neither its minimum nor its maximum at any open interval $I \subseteq \mathbb R$ , then how to prove that $f$ is ...
1
vote
1answer
49 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
2
votes
1answer
34 views

What is the definition of this set of absolutely continuous function

I know that $$AC(a,b):=\left\{f \in C(a,b)|f(x) = f(c)+\int_c^x g(t) d \lambda(t),c \in (a,b), g \in L^1_{\text{loc}}(a,b)\right\}$$ $$AC[a,b]:=\left\{f \in C[a,b]|f(x) = f(c)+\int_a^x g(t) d ...
0
votes
1answer
15 views

In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
0
votes
2answers
40 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
2
votes
2answers
48 views

Requirement for continuity of unit normal vector

When considering a subset $\Omega \subset \mathbb{R}^{n}$. If we consider $\nu$, the outward unit surface normal to $\partial \Omega$, what are the requirements of $\partial \Omega$ which will ...
1
vote
1answer
21 views

Sequence problem dealing with continuity and convergence.

I need help in this question. I figured out a way to solve the question but not sure the proof is valid. This is the question, Given $a \in\mathbb{R}$, and a function ...
1
vote
1answer
38 views

Intuition behind homeomorphism from $B((0, 0), 1) \to \mathbb{R^2}$

In my notes I have that the following function is a homemorphism from $B((0, 0), 1) \to \mathbb{R^2}$ $$h(x, y) \to \frac{f(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} (x, y)$$ where $f = ...
4
votes
4answers
135 views

Real Analysis: Showing $f: \Bbb Q \to \Bbb Q$ is continuous

The following is all working in $\mathbb{Q}$, not $\mathbb{R}$. I am working with the function $f: \mathbb{Q} \to \mathbb{Q}$ defined piece-wise by $f(x)=-1$ if $x^2<2$ $f(x)=1$ if otherwise I ...
-1
votes
1answer
29 views

Proving continuity of Thomae's function at irrational points

Let $h:\mathbb R^+ \to \mathbb R$ be a function such that $h(x)=0$ for every irrational $x$ and for any rational number in $\mathbb R^+$ of the form $\dfrac mn ,$with g.c.d.$(m,n)=1$ , we define ...
3
votes
1answer
45 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, $f_{y}(x)= f(x-y)$, ...
20
votes
4answers
593 views

How does this discontinuity occur in evaluating a nested square root?

This question is based on a comment I made on a question likely to be closed. Let $$y=\sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \dots}}}}}$$ be the classic nested square root which has ...
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.
2
votes
1answer
64 views

Equicontinuity of a pointwise convergent sequence of monotone functions with continuous limit

I was looking at this question, and trying to come up with a counterexample. After thinking about it, I thought the following might be true: Claim: let $\{f_n\}$ be a sequence of continuous, ...
2
votes
2answers
37 views

If $f^{n_o}$ has a fixed point , then does $f$ also has a fixed point , where $f$ is continuous on $\mathbb R$?

In relation to this question , To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$ , if $f: \mathbb R \to \mathbb R $ is a continuous function such that for some $n_o \in \mathbb N$ the ...
1
vote
1answer
43 views

How to show that a complex-valued function is uniformly continuous?

should a function be uniformly continuous in both arguments if it should be uniformly continuous as a complex-valued function. For example how can I proove that ...
0
votes
0answers
25 views

Another question on continuous surjective functions

It is known thta if $f: \mathbb R \to \mathbb R$ is a continuous surjective function that takes every value at most twice then $f$ is strictly monotone . My question is " What is the maximum value of ...
0
votes
2answers
35 views

On continuous surjective function that takes every value at most a finite no. of times

If $f: \mathbb R \to \mathbb R$ is a continuous surjective function which takes every value at most a finite number of times , then is it true that $f$ is strictly monotone ?
0
votes
1answer
18 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 ...
1
vote
1answer
25 views

Derive property from continuity - is this proof valid?

Prove that if $f:R^+ \rightarrow R^+$ is continuous on the positive reals and is decreasing, then for all $a$ there exists an $\eta > 0$ such that $(a-\eta)f(a-\eta) > \frac{1}{2}a*f(a)$. EDIT ...
1
vote
2answers
26 views

If $f: \mathbb{R}^2\to \mathbb{R}$ is Lipschitz, then $g(x)=f(x,a)$ too?

Let $f: \mathbb{R}^2\to \mathbb{R}$ be a Lipschitz continous function, meaning that $ |f(x_1,y_1) - f(x_2,y_2)| \leq L \, || (x_1,y_1) - (x_2,y_2) ||$ for all $(x_i,y_i) \in \mathbb{R}^2$, where ...
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 ...
0
votes
1answer
40 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
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
51 views

How to show $f(x)=\exp((|x|^2-1)^{-1})$ if $|x|<1$ and $f(x)=0$ if $|x|\geq 1$ is a test function?

What would be the formal argument for showing the function $f:\mathbb R^n\longrightarrow \mathbb R$, $$f(x):=\left\{\begin{array}{ccc} ...
0
votes
3answers
61 views

If $f$ is continuous, so is $g=|f|$ [closed]

Prove that if $f$ is continuous, so is $g=|f|$. I need help on this. Thank you. Ok, this is my first time here. The definition of continuity i am using is that $f$ is continuous at $a$ if for any ...
0
votes
1answer
26 views

how to show that $f_n$ is nonnegative on an open interval for all $n$ large enough

Let $\{f_n\}_{n=1}^\infty$ be a sequence of continous functions on $[0,1]$ and for all $x\in [0,1], f_n(x)$ is eventually nonnegative. Show that there is an open interval $I\subseteq[0,1]$ such that ...
0
votes
1answer
23 views

Continuity & boundedness on open interval implies uniform continuity

Suppose f(x) is continuous and bounded on (0,1). Is f(x) uniformly continuous on (0,1)? I think yes, because it's bounded, i.e. there exists $M: |f(x)| < M$. We could use this M as $\delta$ in the ...
0
votes
1answer
44 views

Prove that the inverse image of an open set is open

Let $ X \subset \mathbb{R}$ be a non-empty, open set and let $f: X \rightarrow \mathbb{R}$ be a continuous function. Show that the inverse image of an open set is open under f, i.e. show: If $M ...
1
vote
1answer
109 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
1
vote
3answers
56 views

Show $f$ is uniformly continuous

Let $f$ continuous function on $[0,\infty)$. Lets assume there are $a,b$ such that: $\lim_{x\rightarrow \infty} f(x)-(ax+b) = 0$. Prove $f$ is uniformly continuous on $[0,\infty)$. Well, At ...
0
votes
0answers
15 views

Relation between continuity as a map and joint continuity

Let $f=f(x,y) : \mathbb{R}^2 \to \mathbb{R}$ and denote by $C(\mathbb{R})$ the space of bounded and continuous, real-valued functions on $\mathbb{R}$. Is it true that if the map $x\mapsto f(x,\cdot)$ ...
1
vote
1answer
33 views

uniform continuity on $(a, b]$ implies limit at $a^+$ exists and finite

Let a uniformly continuous function $f$ on $(a, b]$. Prove that $\lim_{x\rightarrow a^+} f(x)$ exists and finite. What I did so far: from the definition of uniform continuity: ...
1
vote
0answers
40 views

Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
0
votes
1answer
19 views

semicontinuity implies sequential semicontinuity

I have that $F:X\to (-\infty,+\infty]$, with $X$ topological space. By definition, $F$ is lower semicontinuous in $x_0 \in X$ if $\forall t \in \mathbb{R}: \. t<F(x_0) \.\exists U\in ...
0
votes
0answers
27 views

Question about a theorem concerning the continuity of integral functions

If we have $$F(t):=\int_V f(t,x)dx$$ where $V$ is some measurable subset of $\mathbb R$ and $x\mapsto f(t,x)$ is a measurable function. Moreover let $F$ be defined for all $t\in U$ a open subset of ...
8
votes
4answers
492 views

Weaker Condition than Differentiability that Implies Continuity

It is a well-known fact that differentiability implies continuity. My question is this: is there some condition for a function that is both weaker than differentiability and stronger than continuity? ...
1
vote
1answer
45 views

Real analysis help: Proof of continuous functions

The question is: Let $h:\mathbb{R}\rightarrow\mathbb{R}$ be continuous on $\mathbb{R}$ satisfying $h(m/2^n)=0$ for all $m\in \mathbb{Z},n\in \mathbb{N}$. Show that $h(x)=0$ for all $x\in \mathbb{R}$. ...
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 $ ...
3
votes
1answer
21 views

$f(x)$ non-decreasing then pseudoinverse of $x + f(x)$ is Lipschitz.

while studying some proof, I came across the following statement: Let $f$ be a non-decreasing function defined on closed interval $[a, b]$. Let $\alpha = a + f(a)$ and $\beta=b+f(b)$. We can ...
1
vote
3answers
44 views

Show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f$ is discontinuous at $c$

How to show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f: \mathbb R \rightarrow \mathbb R$ is discontinuous at $c$ ? I know that $f$ cannot have ...
1
vote
1answer
41 views

Are the two statements about continuous functions equivalent?

I have always wondered about this: A continuous function is defined thus: for any $\epsilon>0$, there exists $\delta\in\Bbb{R}$ such that $|x-y|<\delta\implies |f(x)-f(y)|<\epsilon$ for ...
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 ...
0
votes
2answers
15 views

Technicality: convergent sequence is preserved by continuous map

I'm looking at the proof of a basic result in Maxwell Rosenlicht's analysis book: Let $(E,d)$, $(E',d')$ be metric spaces. Let $f: E \to E'$ be a function. Then if, for every sequence of points ...
1
vote
1answer
42 views

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarrily lipschitz.

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarily lipschitz. Is the above statement true? I thought since $f$ is continuous on a compact metric space, $f$ ...
2
votes
1answer
47 views

Modifications of Weierstrass's continuous, nowhere differentiable functions

Recalling how nowhere continuous functions such as the Dirichlet function can sometimes be modified on a $\lambda$-null set of points (in this instance, a countable set) to become everywhere ...
0
votes
1answer
18 views

Questions regarding regulated functions

I have a few quick questions regarding regulated functions: Firstly, I'll state the definition I have been given: Definition: Let $I=[a,b]$ be a compact interval. Then $f:I\to \mathbb{R}$ is called ...
9
votes
5answers
932 views

Functions that are continuous only at two points?

I need to find a function $f:\mathbb{R}\to\mathbb{R}$ which is continuous only at two points, but discontinuous everywhere else. How on earth would I go about doing this? I can't think of any ...
0
votes
2answers
67 views

Prove $f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ defines a continuous function on $\mathbb{R}$.

Prove $$f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k$$ defines a continuous function on $\mathbb{R}$. I think we can show that if $\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ is uniformly ...
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 ...