# Questions tagged [continuity]

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) or (general-topology)

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### Is it possible to define integration over a discontinous domain?

I'm trying to think about this from the Riemannian integration perspective so let me know if Lebesgue integration or something else is better. An example where I seem to be running into problems is ...
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### Is there a function $f(x)$ that satisfies $f(e^x) = e^x f(x)$?

Does there exist any function $f$ such that $$f(e^x) = e^x f(x)?$$ If so, I believe it could be used to create an analytics versions of the function $\exp_{n+1}(x) = \exp(\exp_{n}(x))$ (where $n$ is ...
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### Differentiable function for $\mathbb{Q} \rightarrow \mathbb{Z}$ [closed]

Are there any continuously (at least once) differentiable function that maps rational numbers $\mathbb{Q}$ to integers $\mathbb{Z}$? How about monotonic? Can you give me an example?
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### show that a set is not closed

Let $$M:=\left\{ x+i \sin\left(\frac{1}{x}\right)\; |\; x \in (0,1) \right\}$$ not closed does not automatically mean open, right? For example, could I use the approach that every limit point of M is ...
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### Does the monotone pointwise limit of continuous functions has left and right limit everywhere?

Let $\{f_n(x)\}_{n\geq 0}$ be a sequence of continuous functions such that $f_n(x) > f_{n + 1}(x)$ for all $n\geq 0$. Assume that $f_n(x)$ converges pointwise to a function $f(x)$. It is clear that ...
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### Let $f: \mathbb R \to \mathbb R$ be a strictly decreasing continuous function.

Let $f: \mathbb R \to \mathbb R$ be a strictly decreasing continuous function. Prove that there exists a real number a such that $f(a) = a$.
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### Find $I_n(a)=\displaystyle \int \limits_0^1 \dfrac{dx}{(x^2+a^2)^n}, \, a\ne0, \, 0\ne n \in \mathbb{N}.$

Find $$I_n(a)=\displaystyle \int \limits_0^1 \dfrac{dx}{(x^2+a^2)^n}, \, a\ne0, \, 0\ne n \in \mathbb{N}.$$ I have following ideas. We have $I_n(a)=I_n(-a)$. \begin{align} \forall \,0\ne n \in \mathbb{...
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### Find the values of $p$ and $q$ to make $f$ continuous

If $$f(x)=\begin{cases} x^2-1&-3\leq x<0\\ px+q&0\leq x\leq 1\\ x+1&1<x\leq 3 \end{cases}$$ is continuous on its domain. Then find the value of $p$ and $q.$ view question
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### How to interpret the categorical relationship between convergent sequences and continuous functions

I attend a math seminar at my university, and a senior professor went on the following (paraphrased) tangent As a side note, there's an interesting relationship between convergent sequences and ...
### Continuity of $\frac{x}{\sqrt{x^2+y^2}}$
I proved that the function $f(x,y) = \frac{x}{\sqrt{x^2+y^2}}$ at $x\neq0$ is continuous and want to extend continuity to $x=0$. Using polar coordinates I found that $\lim_{x,y\to 0} f(x,y)$ is ...
Suppose we have a sequence of $\mathbb{R}^d$-valued stochastic processes $(X^n)_{n \in \mathbb{N}}$ converging to $X$ uniformly on compacts in probability, i.e.  \operatorname{plim}_{n \to \infty} \...