# Tagged Questions

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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### Can $f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $\forall y \in \operatorname{im}(f)$, $f^{-1}(y) = \{a_y,b_y\}$ be continuous?

This is the problem we want to solve: Can $f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $\forall y \in \operatorname{im}(f)$, $f^{-1}(y) = \{a_y,b_y\}, a_y \neq b_y$ be continuous? ...
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### Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
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### If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But let'...
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### Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R$ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
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### A sequence of functions that is uniformly continuous, pointwise equicontinuous, but not uniformly equicontinuous when their domain is noncompact

I'm trying to prove my sequence of functions $(f_n) = \frac{n}{n+1}\cos(x^2)$ on (0,1) is pointwise equicontinuous, uniformly continuous, but not uniformly equicontinuous. But, I'm having a lot of ...
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### 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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### For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
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Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor {x}^{-... 0answers 43 views ### Showing points of continuity of a function f(x) that takes the value 1/n whenever x belongs to a sequence {An} and is zero elsewhere. I am given a sequence (An), n=1,2,3,... which consists of distinct numbers, which converges to 3 as n tends to infinity, but none of its terms are equal to 3. Then I am given a function f(x) =... 0answers 47 views ### Proposed proof of continuous operator on Sobolev space Hi I am interested in a question about continuity: Assume that \Omega \subset \mathbb{R}^{n} is bounded and consider operator$$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow L^{q}...
Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
In Protters book Stochastic Integration and Differential Equations And in uncountable other sources, they mention the continuous sample paths of the brownian motion. That is: It holds that t\...