Articles of continuity

Optimization with a Probability

Imagine two points in $ℝ^2$ at $(-1, 0)$ and $(1, 0)$. You would like to walk from one point to the next in the shortest distance possible. However, there is a line segment coming from the origin to a point $(0, S)$ and $(0,-S)$. In this instance, $S$ is a continuous uniform random variable picked […]

Continuity of inverse function (via sequences)

If $f:[a,b] \rightarrow [c,d]$ is a continuous bijective function, then prove $f^{-1} $ (its inverse function) is continuous on $[c,d]$. I know this can be proven by using monotony of $f$, can anyone help me finish my approach? $\textbf{My approach: }$ Let $s_n \rightarrow s $ on $[c,d]$. And let $ t_n = f^{-1}(s_n),$ and […]

Continuity question: Show that $f(x)=0, \forall x\in\mathbb{R}$.

This question already has an answer here: Can there be two distinct, continuous functions that are equal at all rationals? 4 answers

If $f\circ g$ is continuous and $f$ is a local homeomorphism, then $g$ is continuous

Suppose $g:X\to Y$ and $f:Y\to Z$, and $f$ is a local homeomorphism, which is to say that for any $y\in Y$ there is a neighborhood $U$ of $y$ such that $f\restriction U$ is a homeomorphism from $U$ to $f[U]\subseteq Z$, and suppose also that $h=f\circ g$ is a continuous function. Does it follow that $g$ […]

Thomae's function, doubt in continuous proof in the irrationals.

I was studying about this proof and i almost understand all of it, i just have one doubt there, the proof i found is the following; Let f be defined by; $$ \begin{align} f(x) = \begin{cases} 0 & \text{if $x$ is irrational}\\ \frac{1}{q} & \text{if $x = \frac{p}{q}$ where $(p,q) = 1$ and q > […]

How to evaluate this limit? Riemann Integral

$$\large f(x)= \lim_{n\rightarrow \infty}\left( \dfrac{n^n(x+n)\left( x+\dfrac{n}{2}\right)\left( x+\dfrac{n}{3}\right)… \left( x+\dfrac{n}{n}\right)}{n!(x^2+n^2)\left( x^2+\dfrac{n^2}{4}\right)\left( x^2+\dfrac{n^2}{9}\right)…\left( x^2+\dfrac{n^2}{n^2}\right)}\right)$$ $x\in R^+$ Find the coordinates of the maxima of $f(x)$. My Work: Is the method correct? Is there an easier way?

Definition of sequential continuity: converse?

A function $f: \mathbb R \to \mathbb R$ is called sequentially continuuous if $x_n \to x$ implies $f(x_n) \to f(x)$. Every continuous function is sequentially continuous. Let $f$ be a continuous function on $\mathbb R$. If $y_n = f(x_n)$ is a convergent sequence does it follow that $x_n$ is a convergent sequence? At first I […]

Let $f$ be continuous on the real numbers. Let $c$ be in real number with $f(x)=c$ for all $x$ in $\mathbb{Q}$. Show that $f(x)=c$ on $\mathbb{R}$.

Can someone solve this question? Let $f$ be continuous on $\mathbb{R}$. Let $c$ be in real number with $f(x)=c$ for all $x$ in $\mathbb{Q}$. Show that $f(x)=c$ for all $x$ in $\mathbb{R}$. thank you

If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$?

Let, $id:(X,d_1)\to (X,d_2)$ is continuous. Then which is(/are) TRUE ? (A) $X$ must be singleton. (B) $X$ can be any finite set. (C) $X$ can NOT be infinite (D) $X$ may be infinite but NOT uncountable. My Thought : If $X$ is singleton then the function is continuous trivially but the condition is not necessary. […]

The form $w_p(x,y) = \frac{(x-a)\,dy-(y-b)\,dx}{(x-a)^2+(y-b)^2}$ is exact $\iff$ $\exists$ angle function

Given $p=(a,b)\in\mathbb{R^2}$, show that the $1$-form $w_p:\mathbb{R^2}-\{p\}\to (\mathbb{R^2})^*$, defined by $$w_p(x,y) = \frac{(x-a)\,dy-(y-b)\,dx}{(x-a)^2+(y-b)^2}$$ is closed. Prove that this form is exact in an open $U\subset \mathbb{R^2}-\{p\}$ if and only if there exists a continuous function (necessairly $C^\infty$) $\theta_p:U\to\mathbb{R}$ such that $\cos \theta_p (z) = \frac{x-a}{|z-p|}$ and $\sin \theta_p (z) = \frac{y-b}{|z-p|}$ for all $z=(x,y)\in U$. […]