# 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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### Continuity of rational numbers

Given that a function $f: \Bbb R \to \Bbb R$ is continuous on $\Bbb R$ and that $f(r)=0$ $\forall r \in \Bbb Q$, I need to show that $f(x) = 0$, $\forall x \in \Bbb R$. Can anyone advise on how to ...
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### Proof of Sequential Criterion

Here is my attempt at a proof for the Sequential Criterion. Definition: Sequential Criterion for Continuity A function $f: A \to \Bbb R$ is continuous at the point $c \in A$ if and only if ...
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### Good reference for “solving equation $f(x)=0$ by homotopy and continuation methods”

I need a good reference for "solving equation $f(x)=0$ by homotopy and continuation methods". If $f:X\to Y$ is a continuous map between to linear space $X$ and $Y$, we want to find the roots of $f$. ...
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### Diagnosing essential Classical Mathematical Analysis I knowledge needed for II

I need to take Classical Mathematical Analysis II (Chapters 7-10: Sequences & Series of Functions, Special Functions (Exp/Log/Fourier/Gamma), Functions of Several Variables, Integration of ...
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### Which functions are $C^k$, but not $C^{k+1}$ on $\mathbb{R}$

The only functions I can think of that fulfill this property are the polynomials of degree k. However this does not necessarily imply, that this are the only functions that are in such a space. So I ...
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### Check the differentiablity of $\theta :\mathbb R^2 \setminus\{(0,0)\}\to \mathbb R$

For , $(x,y)\in \mathbb R^2$ with $(x,y)\not =(0,0)$ , let $\theta=\theta(x,y)$ be the unique real number such that $-\pi<\theta \le\pi$ and $(x,y)=(r\cos \theta , r\sin \theta)$ , where ...
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### On existence of continuous mappings mapping open bounded path-connected sets onto open bounded path-connected sets

I was just thinking a bit and this question somehow arrived, I will ask it below after some preparation. Let us deal with this problem in $\mathbb R^n$. Now, suppose that we choose some two subsets ...
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### If the sets $A=\{x\in E, f(x)<\lambda\}$ and $B=\{x\in E, f(x)>\lambda\}$ are open, then $f$ is continuous

I have that $f: (E,\theta)\rightarrow (\mathbb{R},|.|)$ an application, if we have that for all $\lambda\in \mathbb{R}$ the two sets $A=\{x\in E, f(x)<\lambda\}$ and $B=\{x\in E, f(x)>\lambda\}$ ...
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### Working with Ideas related to Continuity

I've been interested in figuring out, if $f^2(x) = f(x) \cdot f(x)$ is continuous, does that confirm that $f$ is continuous? We know that the domain between the two functions are the same, and so for ...
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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 ...
### 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 ...