# 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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### $\varepsilon - \delta$ proof that $f(x) = x^2 - 2$ is continuous - question concerning the initial choice of $\delta$

I just realized I did non really internalized $\varepsilon - \delta$ proofs. Here there is an attempt, with some general questions I have. Proposition: $f(x) := x^2 - 2$ is continuous. ...
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### A discontinuous function on $\mathbb{R}^n$

I have read a paper and I did not understand with the following statement: Let $r_1, r_2,...$ be an enumeration of $\mathbb{Q}^n \subseteq \mathbb{R}^n$. Given functions $g:\mathbb{R} \to \mathbb{R}$ ...
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### Showing Lipschitz continuity

Let $f: \Bbb R^n \to \Bbb R^m$ be a function with the property that for all $v \in \Bbb R^n$ an $L=L(v) \gt 0$ exists so that for all $x \in \Bbb R^n$ the function $t \longmapsto f(x+tv)$ is $L$-...
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### Let $f:R^n\rightarrow R$ be a lower semi-continuity function, how to show for any constant $r$ , $U=\{z\in R^n : f(z)> r\}$ is open?

Let $f:R^n\rightarrow R$ be a lower semi-continuity function, how to show for any constant $r$ $U=\{z\in R^n : f(z)> r\}$ is open ?
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### Obscure proof that $+$ and $\times$ are continuous?

I am looking for proof of $+$ and $\times$ are continuous operations without using the standard definition of continuity (1. $\epsilon-\delta$, or 2. preimage of open sets or 3. sequential ...
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### How to construct continuous functions on arbitrary topological spaces?

Constructing continuous functions $\mathbb{R} \to \mathbb{R}$ (with the euclidean topology) is easy: there is a quite large collection of elementary functions, and we can get even more by composing/...
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### Real Analysis & Continuity

If $f:\mathbb{R} \to \mathbb{R}$ is a continuous function such that $f(x)=x$ has no real solution, then show that $f(f(x))=x$ has no real solution either. Is the proof trivial as it seems or does it ...
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### Uniform Continuity

This question has three parts. a) Difference between continuity and uniform continuity b) Geometrical meaning of uniform continuity c) Correct the example Definition of Continuity of a function ...
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### Continous function $f$

I have a function $f:[a,b]\longrightarrow [c,d]$ that is bijective and monotone increasing. I have to show that f is also a continous function. I wanted to show this by contradiction that for f is not ...
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### Find the value of the constant k that makes the function continuous

h(x) = \begin{cases} x^{2} & \text{if $x\le5$} \\ x+k & \text{if $x>\ 5$} \\ \end{cases} Answer choices are A. k=20 B.k=-5 C. k=5 D. k=30
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### C¹ function in compact and polygonal path connected implies Lipschitz

"Let $f: \Omega \rightarrow \mathbb{R}^{m}$ such that $f \in C^{1}(\Omega).$ Show that, for $K \subset \Omega$ compact and polygonal path connected, $f|_K$ is Lipschitz." I can relate the function ...
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### A discontinuous function with smooth sections

I am searching for $f : U\rightarrow \mathbb R$ defined in an open square $U$ in $\mathbb R^2$ so that $(0,0) \in U$, $f$ is not continuous at $(0,0)$, for each $x$ the function $y\mapsto f(x,y)$ is ...
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### continuous random variable expectation and variance [closed]

you have a continuous random variable X uniformly distributed, and E(X) = 3, calculate the V(X) am stuck, how am i supposed to get the variance with no function in the first place
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### Is the mapping “positive stochastic matrix onto its Perron-projection” continuous?

I am dealing with a topological question concerning the mapping that maps a positive stochastic matrix onto its invariant distribution. I am asking myself if such a mapping is continuous (or ...
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### Show that function $f$ is not continuous in $x=0$ for all $c\in\mathbb{R}$

Show by $\varepsilon-\delta$-criterion that for each $c\in\mathbb{R}$, the function $f\colon\mathbb{R}\to\mathbb{R}$, $$f(x)=\begin{cases}\frac{1}{x}, & x\neq 0\\c, & x=0\end{cases}$$ is not ...
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### If $f \circ f$ continuous prove $f$ continuous

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ strictly increasing so that $f \circ f$ is continuous. Prove $f$ is continuous. I can prove this using sequences, but it's quite tedious. My question is: ...
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### function nondecreasing in both variables, set of discontinuities is a nullset

Let $f\colon [0,1]^2\to\mathbb{R}$ be a function such that $g(x):=f(x,y)$ for any $y$ and $h(y):=f(x,y)$ for any $x$ are nondecreasing functions (the second variable is fixed). Prove that the set of ...
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### a continuous function on $\mathbb{Q}$

Is there a continuous bijective function from $[0,1] \cap \mathbb{Q}$ to $\mathbb{R}$? I think that there is no such function. The set $|[0,1] \cap \mathbb{Q}|$ is countable and $|\mathbb{R}|$ is ...
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### Is Inverse of a function continuous too?

I read an example from "Principles of Mathematical Analysis" by Rudin under the section 'Continuity and Compactness'. According to the example, Let $X$ be the half-open interval $[0,2\pi)$ on the ...
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### If $f:[a,b]\rightarrow R$ is a uniformly continuous function then its absolutely continuous?

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then is it true that $f$ is always absolutely continuous?
Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?