Questions tagged [continuity]

Question 1

Find all the solutions to $f(f(x) + 2020x + y) = f(2021x) + f(y)$ for all $x,y >0$ when: i) $f: \mathbb{N} \mapsto \mathbb{N}$ ii) $f: \mathbb{R}^{+} \mapsto \mathbb{R}^{+}$ The second part was a ...

Question 2

I am working on the following problem from the L’Hôpital’s Rule section of Stewart Calculus, and I would appreciate feedback on whether my approach is sound, as well as clarification on a few things. ...

Question 3

Let $\alpha > 0$ and consider the function $$ f(x,y) = \begin{cases} \dfrac{|\sin(xy) - xy|^{\alpha}}{(x^{2} + y^{2})^{3}} & \text{if } (x,y) \neq (0,0), \\[2mm] 0 & \text{if } (x,y) = (0,...

Question 4

Let $0 \in \Bbb{N}$. Prime number theorists will be familiar with the formula: $$\rho(x) = \pi(x) - \pi(\sqrt{x}) + 1 = \sum_{d \ \mid\ \sqrt{x}\#}\mu(d)\lfloor\frac{x}{d}\rfloor$$ It is usually taken ...

Question 5

NOTE- The source of question is Advanced Calculus on real axis which doesn't cover $\textsf{Lebesgue integral}$ and $\textsf{Measure Theory}$ so therefore an approach by those wouldn't be of much ...

Question 6

I am independently learning calculus from "Calculus, Early Transcendentals" 6th Edition by James Stewart. This pertains to section 5.3 "The Fundamental Theorem of Calculus", ...

Question 7

My teacher used integration by parts to solve the problem like so: $$\int_0^2 xd(\{x\}) =[x\{x\}]_0^2-\int_0^2 \{x\}dx\\ =0-\int_0^1 xdx-\int_1^2 (x-1)dx$$ which comes out to -1. But when I was ...

Question 8

I'm starting my linear algebra studies and came across the following statemtent: $E = F(\mathbb{R};\mathbb{R})$ is the vector space of the one variable real functions $f:\mathbb{R} \rightarrow \...

Question 9

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. For all nowhere-differentiable examples that I know of, for each $a\in\mathbb{R}$ there exist sequences $x_n\to a$ and $y_n\to a$ such that $$\frac{f(...

Question 10

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(x), f(x)+\sqrt{3}, \sqrt{2}-f(x), f(x)+x$ are irrational for all irrational $x$? My attempt: I couldn't come up with any good ...

Question 11

I am looking for literature dealing with topologies on spaces of continuous functions ($C_0$, $C_c$, $C_b$, $\ldots$), particularly with regard to their application when dealing with topologies on ...

Question 12

I have the following exercise and I don't know if my proof is correct: Let $(X,d_{X}),(Y,d_{Y})$ metric spaces with $X=X_{1}\cup X_{2}$ Let $f_{i}:X_{i}\to Y, \ i\in\{1,2\}$ continuous applications ...

Question 13

I'm new to homological algebra. Just wondering why we never seem to see the involved chain complexes defined simply to be $C_n=$ continuous maps "of degree $n$" where the context makes the ...

Question 14

I`m trying to do the following exercise for my general topology class: Let $(X,d)$ a metric space and $B\subset\mathbb{R}^n$ with euclidean metric. Let $f:X\to B$ and application determined by $f_{i}...

Question 15

Let $f:\mathbb{R}\mapsto \mathbb{R}$. The goal is to prove that $f$ is continuous if, and only if, for all $X\subset \mathbb{R}$, $f(\overline{X})\subset \overline{f(X)}$ Let $X\subset \mathbb{R}$, $y\...

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Questions tagged [continuity]

On solving the nested functional equation $f(f(x) + 2020x + y) = f(2021x) + f(y)$

Continuity and Differentiability of $|x|^x$

Analysis of continuity for a two-variable function with a power parameter $\alpha$

Since $\pi(x) - \pi(\sqrt{x}) + 1 = \sum_{d \mid \sqrt{x}\#}\mu(d)\lfloor\frac{x}{d}\rfloor$ is discontinuous, is it everywhere-discontinuous?

Let$\ \ f:[0,1]\to \mathbb{R}$ be a continuous function. Prove $\ \ \lim_{\lambda\to\infty}\int_0^1 f(x)\sin(\lambda x)\,dx = 0$. [duplicate]

Applying the Fundamental Theorem of Calculus to jump discontinuities

Getting different answers for integration problem: $\int_0^2 x d( \{x\} )$

How to prove that $C^k(\mathbb{R})$ is a subspace of $F(\mathbb{R};\mathbb{R})$?

Can there be a continuous function with infinite derivative everywhere?

Does there exist a function such that $f(x), f(x)+\sqrt{3}, \sqrt{2}-f(x), f(x)+x$ are irrational for all irrational $x$?

Topology literature focusing on continuous functions and measures

Continuity of piecewise application between Metric Spaces

Since rings of continuous maps exist why in (co)homology do you never see chain complex $C_n =$ certain ring of continuous functions "of degree $n$"?

$f$ is continuous iff $f_{1},\dots,f_{n}$ continuous in Metric Spaces

Real function is continuous iff image of closure is subset of closure of image [duplicate]

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