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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Proving continuity on spaces of distributions?

Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones. When you have a linear operator ...
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206 views

Is there a continuous nowhere Lanczos generalized differentiable function?

The Lanczos generalized derivative comes from local regression, and is described in this pdf. (You do not need to read that to understand my questions.) If you do read it, observe that I'm pulling ...
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Why is this Takagi's function continuous?

1903 Takagi constructed the function $f: [0,1] \rightarrow \mathbb{R}$ with $f(x) := \sum_{k=0}^\infty 2^{-k} \mathrm{dist}(2^k x, \mathbb{Z})$ where $\mathrm{dist}(x,A) := \inf\{|x-y| : y \in A\}$ ...
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36 views

Intuition on continuty in probability/mean square of a process

How to explain that a process is continuous in probability? I know the definition, but what does it mean? The same with continuity in mean square.
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171 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 ...
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What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
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71 views

A problem of weak* continuity in relation with semigroups

Let $(\Omega,\Sigma,\mu)$ be a probability space. Let $\mathcal{A}$ ba a $\sigma$-subalgebra of $\Sigma$. We denote by $\mathbb{E} \colon L^\infty(\Sigma) \to L^\infty(\mathcal{A})$ the associated ...
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Prove that $f$ is continuous at $(0, y_0)$. where $f$ is defined on $\Bbb R^2$.

Prove that f is continuous at $(0, y_0)$ $f(x, y) = \begin{cases} (1+xy)^{1/x} &\mbox{if } x \neq 0 \\ e^y & \mbox{if } x \equiv 0. \end{cases} $ Thank you!
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44 views

Sufficient conditions for existence of injection from a metric space $M$ to $\mathbb{R}$

Let $M$ be any metric space. What conditions are required of $M$ for there to exist an injective, continuous function $$\varphi \colon M \longrightarrow \mathbb{R}$$ I would like to believe that ...
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60 views

Is $f'$ continuous on $[a,b]$?

If $f$ is continuous on a compact interval $[a,b]$ and has a continuous derivative does it mean that $f'$ is continuous on $[a,b]$ or $(a,b)$?
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29 views

Continuous $L^2$ function has finite sum at integer points?

Let $f\in L^2(\mathbb{R})$ be a continuous function. Is it true that $\sum_{n=1}^\infty |f(n)|^2$ is finite? If the continuity condition is dropped, the statement is not true, because $f(n)$ could ...
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36 views

representation of points of continuity of a function $f :\mathbb{R}\rightarrow \mathbb{R}$

Question is : Suppose $f$ is continuous at $x\in \mathbb{R}$ we need : for given $\epsilon >0 $ existence of $\delta > 0$ such that $|x-y|< \delta$ implies $|f(x)-f(y)|< \epsilon$ ...
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63 views

Equicontinuity and Uniform Boundedness

If we have a sequence of smooth functions $\{f_{n}\}_{n}$ where $f_{n}: U \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$. We are given the following two results: For $x \in U$ we have ...
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238 views

The product of a uniformly continuous function and a bounded continuous function is uniformly continuous

Suppose we have a bounded continuous function $f(x)$ on some interval (a,b). Suppose we also have an function $g(x)$ that is uniformly continuous on the same interval (a,b). Then, is the product ...
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148 views

Functions from the Cantor set

Consider the Cantor set $\Delta\subset [0,1]$. Let $f\colon \Delta\to [0,\infty)$ be a continuous injection. Must $f$ be monotone on some uncountable closed subset of $\Delta$? Note that that van der ...
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Prove the intergral form of Lipschitz continuous

I just want to prove that a function from $\mathbb{R}$ to $\mathbb{R}$ is Lipschitz continuous, if and only if $\exists\, g\in L^p(\mathbb{R}) $ such that $\forall\, x, \, y $, $$f(y)-f(x)\le ...
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88 views

Condition for local absolute continuity to imply uniform continuity

Given that $x\left(t\right)$ is locally absolutely continuous, and $\dot{x}=f\left(x,t\right)$ exists almost everywhere, is it possible to place a condition on $f\left(x,t\right)$ to allow us to show ...
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98 views

On which of the following spaces is every continuous (real-valued) function bounded

On which of the following spaces is every continuous (real-valued) function bounded? i) $X_1 = (0, 1)$; ii) $X_2 = [0,1]$; iii) $X_3 = [0, 1)$; iv) $X_4 =\{t \in [0, 1] : t \mbox{ ...
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19 views

Show that uniform continuity implies stochastic equicontinuity

Let $\Theta$ be a metric space and assume it is compact. Let $W_t: \Omega \rightarrow \mathbb{R}^k$ be a random variable for $t\leq T$. Let $m(.,\theta): \mathbb{R}^k\rightarrow\mathbb{R}^s$. Let ...
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68 views

Convergence of a sequence pointwise a.e. on a manifold given that it converges on a reference manifold

Let $\Gamma_t$ be a compact hypersurface for each $t \in [0,T]$. Let $Q=\Gamma_0\times(0,T)$ and $$Q_T :=\bigcup_{t \in (0,T)} \Gamma_t \times \{t\}.$$ For each $t \in [0,T]$, suppose that ...
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21 views

Applications of Vito Volterra's theorem

We know from Volterra's theorem that: There cannot exist two pointwise discontinuous functions on an interval $(a,b)$ for which the continuity points of one, are the discontinuity points of the ...
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39 views

Intermediate value theorem (IVT) for a function

My teacher said that the function defined by: \begin{equation} f(x)=\begin{cases} \dfrac{1}{x}, & \text{if $x \neq 0$}.\\ 0, & \text{if $x = 0$}. \end{cases} \end{equation} is a ...
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Counterexample to Converse of Extreme Value Theorem?

The extreme value theorem says: If $X$ is a compact topological space, then for all functions $f: X \to \mathbb{R}$ such that $f$ is continuous we have that $f$ satisfies the extreme value property. ...
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59 views

epsilon/delta definition alternative?

The phrase "any epsilon greater than zero" has always seemed somewhat vague. Question: is this an equivalent definition? $\forall\mbox{ Natural Numbers }N>0 \,\,\exists\delta>0\mbox{ s.t. ...
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26 views

Krylov-Bogoliubov theorem without continuity

This question is very closely related to: Continuity in the Krylov-Bogoliubov theorem. The standard counterexample, which is presented in Katok-Hasselblatt is the following: Let $f:[0,1]\rightarrow ...
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58 views

Space of Continuous mappings to metric spaces

I want to ask whether some basic result from the space $C([0,1],R)$, where $R$ is the real space carries over to the space $C([0,1],E)$, where $(E,\|\cdot\|_E)$ is a metric space. We know that ...
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36 views

Continuity of the orthogonal matrix-valued function

Given $d<k$. Suppose that $H:\mathbb{R}^k\rightarrow {\cal M}(\mathbb{R})_{d,k}$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$. Now define a ...
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Continuous function that is invertible in one argument---is its inverse continuous in both arguments?

Suppose that $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous function and that it is invertible in its second argument, i.e. for every $x \in \mathbb{R}$, $f(x,\cdot)$ is invertible with ...
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real analysis continuous definition question

Here is the definition Let $E$ be a subset of $R$ and let $f$ be a real-valued function on $E.$ Then $f$ is continuous on $E$ if and only if $f^{-1}(V)$ is open in $E$ for every open subset $V$ ...
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Intuition behind continuity in topological spaces

I was approaching the following problem: "Let $f \colon X \to Y$ be continuous. Is it true that if $x$ is a limit point of $A \subset X$ then $f(x)$ is a limit point of $f(A)$?" The answer is that ...
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Continuity on open and closed intervals

I will be taking Calculus I soon, and I just want to make sure I understand some concepts correctly. So far, reading my book for Calculus I, I've encountered the definition of continuity as being ...
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How to fulfill those boundary conditions?

The problem is the following: Think of a set of functions depending on spherical coordinates given by: $${\psi}_{l m}(r,\theta,\phi) ={k_l}(ar) P_{l}^{m}(\cos \theta) e^{\pm i m\phi} ,$$ so ...
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How to prove $x^y$ is jointly continuous?

It's known that real exponentiation $x^y$ is continuous in each variable, but is real exponentiation jointly continuous in both the exponent and the base? I considering the function ...
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188 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 ...
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51 views

Discontinuities of monotone operators on arbitrary spaces

Let $X$ be a vector space equipped with an inner product $\langle .,.\rangle$. A function $f:X\rightarrow X$ is said to be monotone if, for all $x,y$, $\langle f(x)-f(y),x-y\rangle\geq 0$. On ...
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70 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 ...
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76 views

Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
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105 views

Extreme Value Theorem and Semicontinuity

Restricting us to function of a single real variable, I was used to prove Extreme value theorem via the short way: show that continuous functions preserve compactness, and the job is done. Now, I ...
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71 views

Presentation of tree decompositions (and related concepts) in terms of continuous maps?

A tree decomposition of a graph $G$ is commonly defined in terms of a tree $T$ with the following structure: Each vertex $t \in V(T)$ is associated to a set $X_t \subseteq V(G)$; The union ...
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130 views

Continuity in set functions

Let a function be defined as $f:(\Omega_1,\mathcal{F}_1)\rightarrow (\Omega_2,\mathcal{F}_2)$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are $\sigma$-fields in $\Omega_1$ and $\Omega_2$ respectively. ...
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For what functions, “$=$” instead of “$\leq$”

Theorem: Suppose $f,g\in C(U,\mathbb R^n)$ and let $f$ be locally Lipschitz-continuous in the second argument, uniformly with repsect to the first. If $x(t)$ and $y(t)$ are respective solutions of the ...
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Continuity of linear form

Let $E=\mathbb{R}[X]$ We define $N:\, P \to \sum_{n=0}^{\infty} { |P^{(n)}(n)|}$ ($P^{(n)}$ being the $n$-th derivative) , it is not hard to prove that $N$ is a norm on $E$. Help me to study the ...
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206 views

Implication of Lipschitz continuity

I am a little bit unsure about the following claim. Let $H:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}$ be a mapping of the form $H(x,f(x))$, where $f:[0,1]\rightarrow \mathbb{R}$ is some continuous ...
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Functions in a Reproducing Kernel Hilbert Space are Lipschitz continuous

I would like to show that all the functions in a Reproducing Kernel Hilbert Space (RKHS) are Lipschitz continuous. So that, I take two points in the domain $\vec{x}_{1} ,\vec{x}_{2} \in X$ then from ...
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Zoo of sigmoid integrals (computational convenience)

In many areas in computational science (e.g. neural networks, fuzzy logic ... ) there is special interest in function like sigmoid ( erf, arctan, tanh ... ) which are kind of blured version of ...
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Is convergence in probability to a uniformly continuous function a sufficient condition for stochastic equicontinuity?

Suppose that a random function $g_T(\theta)$ converges in probability to a function $g(\theta)$ uniformly continuous over $\Theta$ as $T\rightarrow \infty$ $\forall \theta \in \Theta$. Is this ...
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25 views

Is continuity preserved under Max operator and Euclidean norm?

Suppose that the functions $f_i: \mathbb{R}^k \rightarrow \mathbb{R}$ are all continuous over $\mathbb{R}^k$ for $i=1,...,k$. Is the function $$ g(x)=\|\max\{0,f_1(x)\},...,\max\{0,f_k(x)\}\|^2 $$ ...
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Proving equivalent statements of continuity

X and Y are some arbitrary metric spaces and f is a function from X to Y, $f:X \rightarrow Y$. The goal is to prove that these statements are equivalent f is continuous If a sequence ${p_n}$ ...
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Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in ...
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Continuity of a parametric integral (where the integrated function is discontinuous)

For all $t\in\mathbb{R}$ consider $$F(t):=\int_\mathbb{R}e^{-x^2/2}\log|t+e^x|\,dx \;.$$ I managed to show that $F(t)$ is well-defined and finite for every $t$. I would like to show that $F$ is ...