Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) ...

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Given a continuous function with an asymptote, prove that the function is uniformly continuous.

I state the exercise: Given $f: [0, + \infty) \rightarrow R$, f continuous. Prove that if $\lim_{x \rightarrow \infty} f(x) = \lambda$, where $\lambda \in R$ then $f$ is uniformly continuous. My ...
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51 views

Integral of a function and limit

Let $f$ be a positive and continuous function on $[0,1]$. It can be shown there exists, for every natural number $n$, a real natural number $m(n)\in [0,1]$ such that $$ \frac{1}{n}\int_0^1f(x)\,dx= \...
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Find the limit of the cosine sequence. [duplicate]

The question is to find $\lim\limits_{n\to\infty} \cos \frac{x}{2} \cdot \cos \frac{x}{2^2} \cdot \cos \frac{x}{2^3}\cdot\cdot\cdot \cos \frac{x}{2^n}$. I first thought of several facts: 1). $-1 \leq ...
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Why is the multi-variable function $f(x,y)=\exp(-1/(x^2+y^2))$ continuous?

I don't understand this problem: Show that the function $f: \mathbb R^2\to \mathbb R$ given by $$f(x,y)=\exp\left\{\frac{-1}{x^2+y^2}\right\}$$ for $(x,y)≠(0,0)$ and $f(0,0)=0$ is continuous at $(...
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42 views

Which of the following are true? Problem in analysis

Let $f\colon [0,1]\to (0,1)$ be a continuous mapping, then which of the following is not true? 1) if $f(0)< f(1)$ then $f([0,1])$ must be equal to $[f(0),f(1)]$. 2) There must exist $x \in (0,1)$ ...
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93 views

Intermediate value property implies continuity

Let $g\colon \mathbb{R} \to \mathbb{R}$ be a function with the intermediate value property. Let $x \in \mathbb{R}$. Suppose to each sequence $ (x_n) $ converging to $x$ there exists a constant $K$ ...
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1answer
31 views

Norm operators bounded below implies almost uniform lower bound

I have a hard time proving (or disproving) the following statement about continuous linear operators: $$(\exists c>0:\forall j:\|T_j\|\geq c)\Rightarrow(\exists\delta>0:\forall n:\exists x\in ...
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35 views

Homeomorphisms between any two doubly punctured spheres and two punctured $R^n$.

Let $p, q$ be the north pole and the south pole of $S^n$ respectively. Then $S^n-p-q$ is homeomorphic to $S^n-a-b$ where $a,b$, are distinct points in $S^n$. Also $R^n-a$ is homeomorphic to $R^n-b$...
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Example of a continuous surjection from $I=[a,b], a,b \in \mathbb{R}$ to $S^2$.

Continuous surjection from $I=[a,b], a,b \in \mathbb{R}$ to $S^2$. What is an example of such a surjection? I can't think of any. I would greatly appreciate any examples.
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find all values of a and b for which the function will be continuous

I have to find all values of a and b for which the function will be continuous. what I do is next: ...
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0answers
31 views

Non zero continuous path $[0,1]\to \mathbb C$ has continuous logarithm

Let $\gamma:[0,1]\to \mathbb C$ be continuous, and not passing through $0$. How can we prove that, using complex analysis, there is a continuous $G:[0,1]\to \mathbb C$ so that $\gamma=e^G$ ? This can ...
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37 views

Questions on set theory, topological spaces, and continuity.

(My question would not have fit in the title) Questions: a) Let X be a euclidean three dimensional space R$^3$ with the standard Euclidean length and let Y $=\{P_1$such that $P_1=(a_1,b_1,c_1)$and $...
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45 views

Continuous map from topological space $A\rightarrow B$ as composition of quotient map and unique continuous map from quotient space to $B$

Let $(A,T_A)$ be a topological space; $R$ an equivalence relation; $Q=A/R$ the quotient set with $i:A\rightarrow Q$ the quotient map and $T_Q=\{V\subset Q|i^{-1}(V)\text{ open in $X$}\}$ the quotient ...
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2answers
78 views

There cannot exist any non constant continuous function $f: \mathbb{R} \rightarrow \mathbb{Q}$ [duplicate]

Prove that there cannot exist any non constant continuous function $f: \mathbb{R} \rightarrow \mathbb{Q}$. If there exist such an continuous function , it will map interval to a connected subset of $...
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0answers
8 views

Existence of a continuous sum of continuous functions

Suppose we have a function $f:\mathbb{R}^3\rightarrow \mathbb{R}$, which is continuous. Moreover, there exist functions $g,g^*,g^{**}$ such that $f(x,y,z)= g(x,x+y+z)+g^*(y,x+y+z)+g^{**}(z,x+y+z)$. $g,...
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1answer
111 views

Gambler's ruin problem - unsure about the number of rounds

I am quite confused about the following question: "Suppose that (initially) gambler has 10 dollars and adversary has 5 dollars. They repeatedly throw a fair 6-sided die. If numbers 1 or 2 occur, ...
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62 views

Bijective continuous map between $\mathbb{R}$ and $\mathbb{R}^2$

I am currently attending a course on point-set topology and fundamental group. Today we proved in class that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic with their normal Hausdorff topologies....
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3answers
77 views

How is it not the case that every continuous function is uniformly continuous?

What's wrong with my following proof? Suppose $f:\Bbb R\to \Bbb R$ is continuous. Take $x,y\in \Bbb R$. Given $\epsilon >0$ there exists $\delta_1,\delta_2$ such that $$ 0<|x-x_0|<\delta_1 ...
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28 views

Analyticity deduction [duplicate]

If $f$ is a continuous function in the complex plane, then how do I use the fact that if $f^2$ and $f^3$ are analytic on $\mathbb{C}$, then $f$ is analytic on $\mathbb{C}$, where $\mathbb{C}$ is ...
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56 views

Show that if a collection of continuous functions on $X$ separates points, then $X$ must be Hausdorff

I have to show the following: Suppose that $X$ is a topological space for which there is a collection of continuous real-valued functions on $X$ that separates points in $X$. Show that $X$ must ...
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1answer
39 views

Continuous maps on smoth manifolds

Let $M$ be a smooth manifold, $f:M \to \mathbb{R}$ be a $C^{\infty}$ map and $f(p)=0$. **My question:**Does there exist a neighborhood $U$ of $p$ in $M$ such that $f(U)=0$? i know by coordinate ...
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195 views

When is a function of the largest eigenvalue continuous and/or differentiable?

I want to understand why the following function, the largest eigenvalue of a symmetric linear operator, is continuous and Gâteaux differentiable. \begin{equation*} \lambda(V)=\sup_{f \in \ell^2(I):\ \...
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3answers
68 views

Suppose $f_n\rightarrow f$ pointwise. Show that if each $f_n$ is continuous then $f$ is also continuous. [duplicate]

I know that this result is true for uniform convergence, but I'm struggling to find a counter example of such function.
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1answer
25 views

Find the values of A and B to make a continuous function

Let $f$ be a function defined by: $$f(x)= \begin{cases}(2x+1) & \text{if } x<a\\ (x^2+3x-5) & \text{if } a\leq x<b\\ (x-2) & \text{if } x\geq b\end{cases}$$ For which values of ...
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2answers
60 views

Polynomial of odd degree [closed]

How to show that a real valued odd degree polynomial agrees at least at one point with a real valued bounded continuous function??? I seem to be clueless here..please somebody help
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1answer
28 views

Is definite integral a continuous transform on functional space?

Consider the space of all Lipschitz continuous functions on $[0,1]$ equipped with the infinity norm $\|\cdot\|_\infty$. Let $f:[0,1] \to \mathbb{C}$ be Lipschitz. Is $L(f)=\int_{0}^{1}f dx$ continous?...
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Prove that $f=1/\sqrt{x}$ is continuous on the interval $(0,1]$, but not uniformly continuous.

Prove that $f(x)=1/\sqrt{x}$ is continuous on the interval $(0,1]$, but not uniformly continuous. I believe it follows that $f(x)$ is not uniformly continuous because $f(x)$ is not continuous on the ...
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Riemann integrable function not of the following form

Does there exist a Riemann integrable function $f:[a,b]\to\mathbb{R}$ that is not of the form: $$f(x)=\phi(x)+\psi(x)+\sum_{n=1}^\infty a_n 1_{I_n}(x)$$ where $\phi$ is continuous, $\psi$ is zero ...
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30 views

Prove that if $f$ is locally lipschitz of order $\alpha >0$ at $x_0$, then $ f$ is continuous at $x_0$.

We say a function is locally Lipstchitz of order $\alpha$ at $x_0$ if there exists $\epsilon, M>0$ such that $$ |x-x_0|<\epsilon \rightarrow |f(x)-f(x_0)|<M(x-x_0)^\alpha $$ Prove that $f$ ...
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1answer
53 views

$f(x)$ decreasing and positive implies $f'(x)$ converges to 0? [duplicate]

This one should be easy, however for some reason I can't find an easy way to solve it. So if f is a $C^1$ function over $\mathbb{R}$ that is decreasing and positive (so converging to some value, let ...
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1answer
46 views

Continuity of a function and its derivative

Does $f'$ being continuous on $(a,b)$ imply that $f$ is also continuous on $(a,b)$? I think this is probably quite a straightforward question but it's key to solving the problem that I'm working on.
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$f$ is continuous at $x_0$ if and only if $\forall n\in \mathbb{N}$, $a_n<x_0<b_n$, $\lim_{n\to \infty}[f(b_n)-f(a_n)]=0$

This is a question I picked up from Royden's. Let $f$ be an increasing function on an open interval $I$. For $x_0\in I$ show that $f$ is continuous at $x_0$ if and only if there there exists two ...
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1answer
85 views

Proving a sequence of step functions converges pointwise to a function $f$.

Question: Suppose $f: [0,1] \rightarrow \mathbb{R}$ is continuous on $(0,1)$. Prove there is a sequence of step functions $\left\{f_{n}\right\}$ which converges pointwise to $f$ on $[0,1]$. I saw ...
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1answer
23 views

Show $f(x) = x^2$ lipschitz on $[0,1]$

I would like to show this result but I am a bit stuck To show $f(x)$ is lipschitz, show: $$|x^2 - y^2| \leq L |x-y| \quad \forall x,y \in [0,1]$$ Proceed as usual: $|x^2 - y^2| = |x-y||x+y|$ But ...
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1answer
36 views

Does separately continuous on compact set imply boundedness?

Let $f: \overline{\Omega} \times I \rightarrow \mathbb{R}$, is a continuous function on both variables (separately continuous). $\Omega \in \mathbb{R}^n$ is open, bounded. $I$ is a closed interval in $...
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1answer
46 views

Show that $\lim_{r \to 0}\int_{0}^{2\pi}f(r e^{i \varphi})d\varphi = 2 \pi f(0) $

I need to show that $\lim_{r \to 0} \int_{0}^{2 \pi}f(r e^{i \varphi})d \varphi = 2 \pi f(0)$ if $f$ is continuous on a neighborhood of $z = 0$. I was given the following hint: Write $\int_{0}^{2 \...
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1answer
49 views

Prove the sequence of functions converges and the limit function is continuous

Suppose $F :=$ {$f_n: \Bbb R \rightarrow \Bbb R, n=1,2,3,...$} is an equicontinuous family. If the sequence $f_n(q)$ converges for each $q \in \Bbb Q$, show that $f_n(r)$ converges for each $r \in \...
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Can one define an uncountable infinite product? [duplicate]

The continuous version of a sum is commonly called an integral. But what would be the continuous version of a product? The same question in "pictures": $$ \sum \rightarrow \int $$ $$ \prod \rightarrow ...
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What's the difference between uniformly equicontinuous and uniformly continuous?

I am very confused. Thanks in advance. Our definition is that: Uniformly Equicontinuous: $\forall \epsilon>0,\exists\delta>0 \ such \ that \ |s-t|< \delta \ and \ n \in \mathbb{N} \ then \ |...
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1answer
51 views

When is ${{x^2y} \over {(x^2+y^2)^\alpha}}$ continuous, using polar-coordinates

Given $$f ({x,y})= \begin{cases} {{x^2y} \over {(x^2+y^2)^\alpha}},&(x,y) \ne {(0,0)}\\ 0,&(x,y)={(0,0)} \end{cases}$$ For what values of $\alpha$, $f$ is continuous in ${(0,0)}$? I set ...
3
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1answer
57 views

Gaussian process with independent increments

Suppose that we have a continuous Gaussian process $(X_t)_{t \ge 0}$ with independent increments and $X_0=0$. If the increments are also identically distributed, meaning that $X_b-X_a \stackrel{D}{=} ...
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Lower Envelope of a Function is Measurable

Let $f:\mathbb R\to \mathbb{\overline R}$ be arbitrary, then define the lower envelope $$f_*(x):=\lim_{\delta\to 0,\delta> 0}\inf\{f(z)\;|\;z\in(x-\delta,x+\delta)\}$$ We want to show $f_*$ is $\...
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2answers
32 views

Uniform convergence preserves continuity also if the domain is the entire real line?

I have a doubt regarding the result that uniform convergence preserves continuity. Statement: If a sequence of continuous functions $\{f_n\}_n$ $f_n : A → \mathbb{R}$ converges uniformly on $A \...
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1answer
37 views

Continous function from $ \Bbb Q \rightarrow \Bbb R $, $ f = 1 $ for $x > \sqrt2$ and $ f = 0$ for $x < \sqrt2$

I'm not really sure how to go about this problem. Show that $h : \Bbb Q \rightarrow \Bbb R $, with $$ h(x)=\begin{cases} 0 &\text{for $|x|< \sqrt{2}$} \\ 1 &\text{for $|x|>\sqrt{2}$} \...
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1answer
13 views

$\frac{n^{-h} - 1}{h} = -\log n + O(|h|(\log n)^2)$ for $|h|\log n \leq 1$

I'm trying to prove the continuity of $\zeta(s)$. As part of this proof, I've arrived at a term $$ \frac{n^{-h} - 1}{h} $$ which I want to bound. I wanted to see if it was possible to show that this ...
3
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0answers
50 views

Given any path $\phi$, we can find an injective path $\psi$ with the same endpoints as $\phi$ [duplicate]

Suppose $\phi: [0,1] \to \mathbb R^n$ is a continuous map. Does there exist continuous, injective $\psi : [0,1] \to \mathbb R^n$ such that $\psi(0) = \phi(0)$ and $\psi(1) = \phi(1)$ and $Im(\psi) \...
2
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1answer
47 views

The space of continuous fuctions is compact - Other direction!

we all know that if $X$ is a compact topological space then $F(X)$ is compact for all $F\colon X\to \mathbb{R}$ continuous. I was wondering whether the converse is true? For metric spaces I have found ...
0
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1answer
40 views

Finding a transformation that yields a prescribed PDF

I am attempted to procure a function from a composition when given the PDF (I typed the full problem at the bottom in its entirety in case I left out details in my inquiry). I understand how to get ...
0
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1answer
55 views

Example of function $f:\Bbb R\to \Bbb R$ continuous at $x$ iff $x$ is trascendental [duplicate]

I've seen examples of functions that are continuous only at the irrationals, and discontinuous at the rationals, and I wanted to go a bit further, but I couldn't come up with a function like the one ...
0
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1answer
36 views

Show that there exists a partition $-\infty=t_0<t_1<…<t_k=\infty$ such that $\lim_{t\rightarrow t_j^{-}} F(t)-F(t_{j-1})<\epsilon$

Consider a real-valued random variables $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with cumulative distribution function $F(t):=\mathbb{P}(X\leq t)$. I want to show that ...