Articles of continuity

$X$ a connected space ; if all continuous functions from $X$ to $\Bbb R$ are uniformly continuous, then X is a compact

Let $X$ be a subset of a normed vector space. $X$ is also a connected space. Show that if all continuous functions from $X$ to $\Bbb R$ are uniformly continuous, then X is a compact. Need some help ; thank you 🙂

$f$ is continuous and $f(V)$ is open whenever $V$ is open $\implies$ $f$ is monotone

Let $ A $ be a non-empty subset of $\mathbb R$ and $f : A \to \mathbb R$ be a continuous function on $A$ such that $f(V)$ is an open set for any open set $V$ , then how to prove that $f$ is monotone ?

Continued matrices-valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$. I’d like to construct a continuous function $K:\mathbb{R}^k\rightarrow {\cal M}_{k\times (k-d)}(\mathbb{R})$ such that $K(x)$ is full rank […]

Showing the continuity of $d(x,f(x))$

Assume that $(X,d)$ is compact, and that $f: X \to X$ is continuous. Show that the function $g(x) = d(x,f(x))$ is continuous and has a minimum point. Consider the function $g(x) = d(x,f(x))$. If $g$ is continuous, then $\forall \epsilon >0 \ \exists \delta > 0$ such that $$d(x,y) < \delta \implies d(g(x), g(y)) < […]

Does a function have to be bounded to be uniformly continuous?

My book defines uniform continuity as a form of continuity that works for any points $a$ and $x$ in an interval $I$ such that $$|x-a| < \delta$$ implies that $$f(x) – f(a) < \epsilon$$ It then goes on to assert that “If $f$ is continuous over a closed and bounded interval $[a,b]$, it is uniformly […]

Continuity, uniform continuity and preservation of Cauchy sequences in metric spaces.

Let $ X $ and $ Y $ be metric spaces, and let $ f: X \to Y $ be a mapping. Determine which of the following statements is/are true. a. If $ f $ is uniformly continuous, then the image of every Cauchy sequence in $ X $ is a Cauchy sequence in $ […]

Is this Epsilon-Delta approach to prove that $e^x$ is continuous correct?

I couldn’t find an epsilon-delta proof for continuity of $e^x$ so here’s my take: Suppose $|x – x_0| < \delta$ and fix $\epsilon >0$ Consider $|e^x – e^{x_0}| < \epsilon$, then \begin{gather} -\epsilon < e^x – e^{x_0} < \epsilon \\ e^{x_0} – \epsilon < e^x < e^{x_0} + \epsilon \\ ln(e^{x_0} – \epsilon) < x […]

Is the integral over a component of a doubly continuous function continuous?

Given a continuous function $$f:(0,1)\times(0,1)\to\Bbb R$$ so that $t\mapsto f(t,x)$ is integrable for all $x$, does it follow that $$x\mapsto\int_0^1 f(t,x)\,dt$$ is continuous in $x$? I would think not necessarily, in part because we are considering open intervals which blocks trivial proofs using uniform continuity. But I cannot think of a counter example.

A semicontinuous function discontinuous at an uncountable number of points?

This is an exercise from A second course on real functions. Do you have an example of a semicontinuous function defined on $[0,1]$ which is discontinuous at an uncountable number of points?

$C ( \times \to \mathbb R)$ dense in $C ( \rightarrow L^{2} ( \to \mathbb R))$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense in $L^{2} ([0,1] \to \mathbb R)$ (the usual Lebesgue space). Now consider the space of continuous functions on $[1,2]$ taking […]