# 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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### $S$ be $\pi$-system on a set, given two measures on $\sigma(S)$, is there a topology on $\sigma(S)$ making $S$ dense, and the two measures continuous?

Let $\Omega$ be a non-empty set , $S \subseteq \mathcal P(\Omega)$ be a Pi system (https://en.wikipedia.org/wiki/Pi_system ) on $\Omega$ , let $\sigma(S)$ be the $\sigma$-algebra generated by $S$ (i.e....
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### How to find E(x) and Var(x) in this specific continuous probability distribution.

I've got into some confusion on continuous probability distributions, and everything related to it. This is the problem: Problem Image. I assume from the sketch that pdf is $f(x) = x$ for values of $x$...
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### Continuity of a function on a square

Fix some $\ell\in\mathbb{R}^+$, and say I have a function $f:[0,\ell]\times[0,\ell]\to\mathbb{R}$ with the following properties: $f(s,t)$ is continuous everywhere except when $s=t$, where it is ...
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### How to show continuity of a function with $n-1$ exponentiations?

Say we are given a function $$\Gamma(x)=f_1(x)^{f_2(x)^{\cdot^{\cdot^{f_n(x)}}}}$$ where $f_i,i\in[1;n]$, are continuous functions in their domains. Also assume that the function makes sense, e.g., ...
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### Continuous functions and metric topology [on hold]

Let $X = C[0, 1]$ be the set of all continuous functions $f : [0, 1] \to \Bbb{R}$ (where the domain and codomain have their usual topologies). Let $d_1 : X \times X \to \Bbb{R}$ be the metric on $X$ ...
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### A problem about a continuous iterated function [duplicate]

Let $f:\mathbb {R} \rightarrow \mathbb { R }$ be a continuous function such that $f\circ f \circ f=\text{id}_\mathbb{R}$. Show that $f=\text{id}_\mathbb{R}$. Is there any hint to prove this? ...
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### Limits, and Continuity - Finding whether a function is continuous or not

$$f(x) = \lim_{n \to \infty} \frac{\log(2 + x) - x^{2n}\sin x}{1 + x^{2n}}$$ then, check the continuity of the function at $x = 1$. I found this question in a text. After some thinking I ...
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### Does the limit exist for $y=\sqrt{x}\sin \frac{1}{x}$?

For the graph of $$y=\sqrt{x}\sin \frac{1}{x},$$ Do the following limits exist? If so, what is it? (a) $\lim_{x \to 0^+} f(x)$ (b) $\lim_{x \to 0^-}f(x)$ (c) $\lim_{x \to 0}f(x)$ By the way, the ...
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### Continuity of a function from a pretopological space

It is known that for a function $f$ from a topological space to interval $[0;1]$ to be continuous, it is enough that preimages $f^{-1}]a;1]$ and $f^{-1}[1;a[$ be open in $[0;1]$ for every $a$ in our ...
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### Maps of profinite sets

I was trying to prove (or disprove) whether or not all maps between profinite spaces are continuous. One proof in favor was the following. Suppose we have a map of profinite sets $X\to Y$. For any ...