Continuous Extension Theorems in Real Analysis Stackexchange
Questions tagged [continuity]
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) or (general-topology)
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Linear but discontinuous functions
Consider $\mathcal C^0$ = {$f : (0, 1) → \mathbb R$ | f is continuous} and $E_{\beta}(f) = f(\beta)$, for $\beta \in (0,1)$, and the metric $$ d(f,g) = ||f-g||_2 = \sqrt{\int_0^1(f(x) - g(x))^2dx}$$. ...
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Can be a continuous function at $x=a$, but doesn't have limit at $x=a$? [duplicate]
I know, from calculus the definition of continuity is given below: $$\lim_{x \to a} f(x) = f(a).$$ So most of cases, if $f(x)$ continuous at some point, the limit also exists. However, I have a '...
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4 votes
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Are real numbers only an approximation of a continuous line?
It is clear for me that the $\mathbb R$ set is infinite, but I cannot understand that it is continuous (I am an engineer with formal education in calculus, algebra, geometry, discrete maths and others ...
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-3 votes
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Is a homogeneous of degree one function uniformly continuous on the nonnegative orthant?
Take a function $f:R^N_+ \to R$, where $R^N_+ = \{x \in R^N : x_i \geq 0 \text{ for all } i=1,\ldots,N\}$. Assume $f$ is homogeneous of degree 1, i.e., $f(\alpha x) = \alpha f(x)$ for all $\alpha \geq ...
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What is $\max_{x\in\mathbb R}\int_0^T f(t)f(t+x)\,dt$ with $f\in C^0(\mathbb R)$ and $T$ periodic?
What is $\max_{x\in\mathbb R}\int_0^T f(t)f(t+x)\,dt$ with $f\in C^0(\mathbb R)$ and $T$ periodic ? I tried with some continuous functions and I feel that the value of the max is $\int_0^T f(t)^2\,dt$....
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Finiteness assumption of Lusin's Theorem
In Folland, for example, theorem is given as follows. Suppose $\mu$ is a Radon measure on $X$ and $f : X \to \mathbb{C}$ is a measurable function vanishing outside a set of finite measure. Then for ...
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3 votes
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$f$ satisfies $f(x+y)=f(x)+f(y)$ and is continuous in $0$. Prove it is continuous on $\mathbb R$ [duplicate]
Let $f:\mathbb R \to \mathbb R$ be a continuous function in $0$ such that $$f(x+y)=f(x)+f(y) \quad\forall x,y\in \mathbb R$$ Prove that $f$ is continuous on $\mathbb R$. I'm not asking for the proof ...
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Extend a $C^2$ function from the Boundary to the Interior while Preserving Some Regularity
Let $U$ be some open connected bounded set in $\mathbb{R}^n$ with $C^2$ boundary and $g \in C^2(\partial U)$. Can we extend $g$ to the interior of $U$ such that ${extension}(g) \in W^{2, 2}(U)$? I am ...
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Continuity of minimum function on product spaces
Let $X, Y$ be topological spaces, and $Y$ is compact, and let $f: X \times Y \rightarrow \mathbb{R}$ be a continuous function. Define $g: X \rightarrow \mathbb{R}$ as $g(x) = \inf_{y \in Y} f(x,y)$. ...
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Density of a subspace of continous functions for the norm $1$.
Let $(E,\|\cdot \|_1)$ be the normed space of continuous functions from $[0,1]$ to $\mathbb R$. Let's consider $F$ the subset of $E$ of functions verifying $f(0)=0$. I want to show that $F$ is dense ...
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1 vote
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Does the extension of uniformly continuous functions uniformly continuous? [duplicate]
I already know that if $E\subset\mathbb{R}$, $E_{1}$ is a dense subset of $E$, and there is a uniformly continuous function $f_{1}(x)$ on $E_{1}$, then there is a unique function $f(x)$ such that $f(x)...
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2 votes
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Stochastic matrix is a continuous linear transformation
I try to solve the following two exercises: Notice that $A$ is an $N$ by $N$ matrix and $x$ is a vector in $\mathbb{R^n}$. I want to show that the linear transformation $$ A x=x$$ is continuous. ...
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given $xf''(x) + f'(x)+f(x)\leq 0\,\forall x > 0.$ Find $\lim\limits_{x\to\infty} f(x)$.
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a twice-differentiable function so that $xf''(x) + f'(x)+f(x)\leq 0\,\forall x > 0.$ Find $\lim\limits_{x\to\infty} f(x)$. It may be useful to consider a ...
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Showing that for all $n\ge1$, there exists $α\in\mathbb{R}^+$ and $c\in(a,b)$ so that $\sum\limits_{i=0}^nf(c+iα)=(n+1)(c+\frac n2α)$
Let $f:[a,b]\to (a,b)$ be a continuous function. Show that for all $n\ge 1,\exists \alpha \in\mathbb{R}^+$ and $c\in (a,b)$ so that $\sum\limits_{i=0}^n f(c+i\alpha)= (n+1)(c+\dfrac{n}2 \alpha)$. ...
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1 vote
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What is $\int_x^{2x}f(t)/t\,dt$? [duplicate]
Let $f\in C^0(\mathbb R)$, What is : $$\lim_{x\to 0}\int_x^{2x}\dfrac{f(t)}{t}\,dt$$ I found that : $\forall x\in\mathbb R, \int_x^{2x}f(t)/2x\,dt\le\int_x^{2x}f(t)/t\,dt \le \int_x^{2x}f(t)/x\,dt$. (...
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