# 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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### Evolution of Definitions

I try to understand how the definitions of mathematics have evolved (or formulated)... I'll use the epsilon-delta continuity definition as an example to ask my question... It may seem trivial, but ...
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### The limit of $(\sqrt{1+kx}-\sqrt{1- kx})/x$ as $x\to 0$ [closed]

For what value of k, $$f(x)=\begin{cases}\frac{\sqrt{1+kx}-\sqrt{1- kx}}{x} & \mbox{ if }-1 \le x <0 \\ \frac{2x+1}{x-1} & \mbox{ if } 0\le x<1\end{cases}$$ is continuous at $x= 0$.
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### Function continuity problem

Let f be a continuous function in the interval $[a,\infty)$, and $\lim_{x \to \infty}f(x)=L$. Then f is bounded. I've been trying to prove by contradiction but I couldn't manage to prove that if it's ...
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### Is breaks in the continuity of a graph scalar with regards to multiplying a constant?

I think I'm probably right but seeing how I've recently begun to go down the rabbit hole in mathematics that tends to produce all sorts of inconsistenties with any advanced conjectures that I make, I ...
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### Proving Continuity On A Particular Function

I'm working on a problem where $S \subset \mathbb{R}$ arbitrary and I have a function $f(x) = \inf\{|x-s| : s \in S\}.$ I want to show that $f$ is uniformly continuous for all $x \in \mathbb{R}.$ I am ...
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### Continuity properties of an example function $f:\mathbb{R}^n\to\mathbb{R}$

Consider the function $f:\mathbb{R}^n\to\mathbb{R}$ defined as follows: $$f(x)=\begin{cases} ||x||^2 & \text{if ||x||\le 1,}\\ 1/||x||^2 & \text{if ||x||> 1,} \end{cases}$$ where ...
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### Contractibility is a Weaker Notion than Deformation Retract to a Point [duplicate]

This is problem 6 in Chapter 0 of Hatcher's Algebraic Topology. Let $X$ be the subspace of $\mathbf R^2$ consisting of the horizontal line segment $[0, 1]\times \{0\}$ together with the vertical ...
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### Is the second derivative of $\frac{1}{|x-y|}$ in $L^2(\mathbb{R}^3)$ or $L^1$?

I was unsuccessfully trying to show whether the function $\frac{\partial^2 }{\partial x_k \partial x_j}\frac{1}{|x-y|}$ for $x,y\in \mathbb{R}^3$ is in $L^2$ or in $L^1$? i.e if ...
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### Continuity of a function from the reals to the reals that fixes rational numbers

Define $f:\mathbb{R} \to\mathbb{R}$ by $x\mapsto x$ if $x$ is rational and $x\mapsto 0$ if $x$ is irratoinal. Prove that $f$ is continuous at a point $a\in\mathbb{R}$ if and only if $a=0$. I'm ...
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### if the inverse images of all closed balls are closed, is $f$ continuous?

Is the following statement true? (it is asked to be proved true) If $f: D \to\mathbb R^n$, and for every closed balls $B$ in $\mathbb R^n$, pre-image of $f$ of $B$ is closed in $D$, then $f$ is ...
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### Example of a function differentiable in a point, but not continuous in a neighborhood of the point?

Is there a function that is differentiable in a point $x_0$ (and so continuous of course in $x_0$) but not continuous in a neighborhood of $x_0$ (as said, besides the point $x_0$ itself)? Can anyone ...
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### On preimage of open sets of functions on real line having at most countably many discontinuity points

Let $f:\mathbb R \to \mathbb R$ be a function whose set of discontinuity points is at most countable ; is it true that for every open set $G \subseteq \mathbb R$ , there is an open set $U$ and a ...
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### Local extrema proof by contradiction?

I have a continuous function $f$ over an interval $\left [ a,b \right ]$ such that $f(a)=f(b)$. Also $f$ admits no local extrema over this interval. I can say that this function reaches a maximum ...
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### Classifing a singularity.

I have the function: $$f(z)=z \cos\left(\frac{1}{z}\right)$$ which has a singularity at $z=0$ and $f(z)\to 0$ as $z\to0$. The theory says that a limit implies that the singularity is removable. But ...
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Suppose $f(x,y)$ is jointly continuous and strictly increasing in both $x$ and $y$. For fixed $y$, let $f_y^{-1}$ be the inverse of $f(\cdot,y)$. How can I show that, for a fixed $x$, the function y ... 1answer 17 views ### Polar coordinates approach to find continuity at a point I have to investigate continuity at (1,2) of the following function:- $$f(x,y)= x^2+2y, (x,y)\ne(1,2) ;$$ $$f(x,y)=0 ,(x,y)=(1,2)$$ My approach:- I have considered a circle of radius r around ... 1answer 20 views ### What method can I use to determine continuity of squares on a 2d grid? I have N squares aligned to a 2d grid. I'd like to know if the set is continuous -- that is to say, that each of the N squares is adjacent to at least one other square in the set. This is quite ... 1answer 197 views ### Countable-infinity-to-one function Are there continuous functionsf:I\to I$such that$f^{-1}(\{x\})$is countably infinite for every$x$? Here,$I=[0,1]$. The question "Infinity-to-one function" answers is similar but without the ... 0answers 48 views ### Visit probability as a function of continuous time I am working on a project aiming to model visit probabilities in spacetime prisms. On a given location, I know the visit probability at any time (within the prism boundaries), i.e. the visit ... 1answer 411 views ### Infinity-to-one function Are there continuous functions$f:I\to S^2$such that$f^{-1}(\{x\})$is infinite for every$x\in S^2$? Here,$I=[0,1]$and$S^2$is the unit sphere. I have no idea how to do this. Note: This is ... 1answer 27 views ### Verification of proof of continuity between metric spaces and deduction from proof Let$M = [0,1]^{[0,1]}$and$d(f,g) = \sup{\{\lvert f(x) - g(x)\rvert \mid x \in [0,1]\}}$. For$a,b \in [0,1]$let$\phi_{a,b}(f) = f(b) - f(a)$($\phi$maps from$M \to \Bbb{R}$). Assume that ... 0answers 36 views ### Prove that$\phi : C[a,b]\rightarrow \mathbb{R}$given by$\phi(f)=\int_a^bfdx$is uniformly continuous Prove that$\phi : C[a,b]\rightarrow \mathbb{R}$given by$\phi(f)=\int_a^bfdx$is uniformly continuous. First, since$f$is continuous$\phi$is well defined (integrals exist). Since$f$is ... 1answer 54 views ###$X,Y$be connected ;$f:X\to Y$be a continuous function which is right-cancellative w.r.t. continuous maps on connected spaces ; is$f$surjective? [closed] Let$X,Y$be connected topological spaces and$f:X\to Y$be a continuous function such that for any connected space$Z$and any continuous functions$g_1,g_2:Y \to Z$,$g_1 \circ f=g_2 \circ f ...
Where is the given function discontinuous? f(x)= \begin{cases} x^2&\quad\text{if }x < -1\\ \sqrt{x + 4}&\quad\text{if }−1 ≤ x ≤ 0\\ \sin(2x)/x&\quad\text{if }x > 0\\ ...
My question is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous map such as each irrational number is mapped to a rational number (i.e. ...