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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An open ball is an open set by the continuity of the norm

I Hvev an exercise that I should prove that in a normed space, an open ball is an open set, but using the property of the continuity of the norm. Given a normed space $X$ and the norm $||.||$ such as ...
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66 views

Does the convergence of $\int\limits_0^\infty f(x) dx$ imply $\lim\limits_{x \to \infty } xf(x) =0$?

Let $f:[0, \infty) \to \mathbb{R}$ be a continuous function. Does the convergence of $\int\limits_0^\infty f(x) dx$ allways imply $\lim\limits_{x \to \infty } xf(x) =0$?
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How does this argument show continuity?

I want to show that the unitary group $U(\mathcal H)$ of a Hilbertspace $\mathcal H$ is a topological group wrt the strong operator topology. For the standard proof it is most convenient to use that ...
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22 views

Find all real and continuous functions that are a 3-involution.

Find all continuous functions $f: \mathbb R \rightarrow \mathbb R$such that $f(f(f(x)))=x$. Obviously one solution to this functional equation is $f(x)=x$. If the function is NOT continuous, there ...
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2answers
33 views

Continuity of $1/f(x)$

Suppose that $f(x)$ is continuous on $[a, b]$ and that $f(x) \geq c > 0$ for some constant $c$. Prove that $1/f(x)$ is continuous on $[a, b]$.
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Let $f$ be an continuous function and show that it exists. [closed]

I need some help with this question: Let $f:[0,1] \rightarrow [0,1]$ be a continuous function. Show that there exists a $c \in [0,1]$ such that $f(c)^2 = c$. Any help will be really appreciated. ...
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42 views

Continuity of two variable function

I've got the following exercise and I need some help trying to figure out what's a proper way of proving it: Take the function $$f(x,y)= \begin{cases} \sqrt{1-x^2-y^2}, & \text{if $x^2+y^2<1$} ...
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55 views

Prove $X$ is connected $\iff$ for every continuous $f:X\rightarrow \mathbb R$, $f(X)$ is connected and $ \subset \mathbb R$

Prove $X$ (Metric Space) is connected $\iff$ for every continuous $f:X\rightarrow \mathbb R$, $f(X)$ is connected and $ \subset \mathbb R$ I'd appreciate if somebody posted a whole answer, so that I ...
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39 views

Why does this function not satisfying the definition of continuity?

Let $f: [0,\infty) \rightarrow \Bbb R$, $f=x^{1/2}$, $f$ is continuous. But if $S=\Bbb R$, then $f^{-1}(S)=[0,\infty)$, this is saying the preimage of open set is not open, which seems to contradict ...
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47 views

$\Phi : \mathbb R \rightarrow \mathbb R$ , $f$ is R.I. $\rightarrow \Phi \circ f$ is R.I.

$\Phi : \mathbb R \rightarrow \mathbb R$ is continuous and $\Phi(0)=0$. Suppose that $f$ is Riemann Integrable. Show that $\Phi \circ f$ is also R.I. I assumed $f$ has bounded support $[a,b]$. Thus $...
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39 views

Can we construct a continuous but nowhere differentiable surface?

In my analysis course we learned about the Weierstrass function which is continuous but nowhere differentiable, is it possible to make a surface which is continuous and nowhere differentiable?
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69 views

$X,Y$ infinite dimensional NLS , not both Banach , then $\exists T \in \mathcal L(X,Y)$ such that $R(T)$ is not closed in $Y$?

Let $X,Y$ be infinite dimensional normed-linear spaces , not both Banach , then does there necessarily exist a continuous linear transformation $T:X \to Y $ such that $range (T)$ is not closed in $Y$ ?...
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77 views

To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous

Show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous. Suppose $\{f_n\}$ is a sequence of lower semicontinuous functions on a topological space $X$. ...
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21 views

Convergence in the product of spaces of iteratively composed functions.

My question is a bit odd, in fact conceptually it is not difficult, only that it operates on objects that are complex (to me). I would like to check two types of convergence in the product of the ...
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2answers
106 views

Let f:[0,1]→[0,1] be a continuous function. Show that there exists c ∈[0,1]. Such that f(c)^2 = c. [closed]

Let $f:[0,1] \rightarrow [0,1]$ be a continuous function. Show that there exists a $c \in [0,1]$ such that $f(c)^2 = c$. This all the information I have. I am not quite sure of what to do.
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Does the new definition imply continuity and vice versa?

Let $f: X \rightarrow Y$ in metric spaces, and let $x^*\in X$, and $y^* = f(x^*)$, $f$ is continuous at $x^*$ iff $\forall \epsilon \gt 0, \exists \delta \gt 0$ such that $B_{\delta}(x^*) \subseteq f^{...
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48 views

Discontinuity under uniform convergence

Suppose I have sequence of functions $\; f_n : [a, b] → \mathbb R$ that uniformly converges, i.e. $f_n \rightrightarrows f$ as $n \to \infty,$ and $f_n$ has finitely many discontinuities, $\forall\; n ...
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1answer
33 views

Compute Limit Involving Integral and Periodic function [closed]

If $f: \bf R \to \bf R$ is continuous and periodic with period $T$, then show that $$ \frac{1}{t}\int_{a}^{a+t}f(s)ds \to \frac {1}{T}\int_{0}^{T}f(s)ds$$ where $a\in \mathbb{R}$ and $ t \to \infty$...
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29 views

Switching the quantifier of definition for Continuity

Let $f: X \rightarrow Y$ in metric spaces, and let $x^*\in X$, and $y^* = f(x^*)$, $f$ is continuous at $x^*$ iff $\forall \epsilon \gt 0, \exists \delta \gt 0$ such that $B_{\delta}(x^*) \subseteq f^{...
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36 views

Finding limits of functions by considering factorizations

I had a go at these and ended up writing way too much. Would a quicker way be to find the limits by considering factorizations in this case?
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41 views

Let f be increasing and bounded above. Show that the limit of f when x tends to infinity exists as a real number.

I tried considering the Mean Value Theorem but haven't gotten far. How could I prove that the limit exists as a real number?
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Let $f:\mathbb R\to \mathbb R$ be continuous, $f(x)\to+\infty$ as $x\to\pm\infty$. Show that $f$ has a minimum. [duplicate]

I am not sure how to prove this. Although could a proof of even-degree polynomials with positive leading coefficient be helpful in this case? Let $f:\mathbb R\to \mathbb R$ be continuous and such ...
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Given $f_n(x) = \left(\frac{x}{x+1}\right)^n \sin(x), x > 0$, is it uniformly convergent, pointwise equicontinuous, uniformly equicontinuous?

I have this question I have been cracking for hours I need help. Preface: not a homework question, just trying to gain some intuition on this equicontinuous property. Given $f_n(x) = (\frac{x}{x+1}...
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61 views

What is the most painless way to check whether a $f: \mathbb{R}^n \to \mathbb{R}^n$ and $f : \mathbb{R}^n \to \mathbb{R}^{n\times m}$ is continuous

Given $f(x_1,x_2) = \begin{bmatrix} x_1^2 + x_2 \\ x_1 + x_2^2\end{bmatrix}$ and $g(x_1,x_2) = \begin{bmatrix} 2x_1 & 1 \\ 1 & 2 x_2\end{bmatrix}$ In my class I am only taught the $\...
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Does Lp-convergence and uniform boundedness in $C^2$, imply $C^{1}$ convergence?

Take a sequence $f^{n}$ in $C^{2}([0,1])$, the space of twice continuously differentiable functions, such that $f^{n} \rightarrow f$ in $L^{p}([0,1])$ (the Lebesgue space) for a $f \in L^{p}([0,1])$ ...
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Show that $\sqrt{x^2+x+2}$ is defined and continuous

Show that the function $g(x)=\sqrt{x^2+x+2}$ is defined and is continuous on $\mathbb{R}$. I have tried completion of square for $$x^2+x+2=\left(x+\frac{1}{2}\right)^2+\frac{7}{4}$$ This means ...
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45 views

$f$ is uniformly continuous on $[a,b]$ and $[b-1,c]$ $\Rightarrow$ uniformly continuous on $[a,c]$

$f$ is uniformly continuous on $[a,b]$ and $[b-1,c]$ $\Rightarrow$ uniformly continuous on $[a,c]$ My thoughts: Without loss of generality we only need to show that it's uniform if $x_1 \in [a,b]$...
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$f:[a,b] \to \mathbb R$ continuous a.e. , is there a sequence of functions of finite discontinuity on $[a,b]$ converging uniformly to $f$ on $[a,b]$?

Let $f:[a,b] \to \mathbb R$ be a function continuous almost everywhere , then does there exist a sequence of functions $\{f_n\}$ on $[a,b]$ , where each $f_n$ has at most finitely many discontinuity ...
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Some details in the proof to show that a reciprocal function is continuous.

Let $(X,d)$ be a metric space and $f:X\to\mathbb{R}$ continuous. Let $Y=\{x\in X:f(x)\neq0\}$. Prove in detail that the function $g:Y\to\mathbb{R}$ defined by $g(x)=\frac{1}{f(x)}$ is continuous. ...
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Can $f \in C[0, 1]$ with a countable number of zeros be a zero divisor?

Let us consider the ring of all continuous functions $ C[0,1]$. Let $f \in C[0,1]$ be such that $f$ has countable number of zeros. Can $f$ be a zero divisor? Surely $f$ can't be an unit. If it ...
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Proving $\frac{x^3y}{x^2+y^2}$ is continuous.

Consider the function $g:\mathbb{R^2} \rightarrow \mathbb{R}$ defined by $g(x,y) = \left\{\begin{matrix} \frac{x^3y}{x^2+y^2}& \textrm{if } (x,y) \neq (0,0) \\ 0 & \textrm{if }(x,y) = (0,...
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Function that is continuous only on the transcendental numbers

Recently I saw Thomae's Function, also called the Popcorn Function, which is continuous at the irrationals and discontinuous at the rationals. This made me wonder: is there a function which is ...
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70 views

Covering Space Counter example

Given a covering space $(p,\tilde{X})$ of a space $X$, we know that every covering application $p:\tilde{X} \rightarrow X$ is a local homeomorphism and possesses the $\textbf{unique path lifting}$ ...
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Proving that a piecewise function $f(x,y,z)$ cannot be continuous at the origin

I am presented with the following problem. Consider the function $f:\mathbb{R}^3 \to \mathbb{R}$ defined by $$f(x,y,z)= \begin{cases} \frac{z}{x}+y & \text{if} \: x \neq 0, \\ a & \...
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Hölder continuous function in one of the multiple variables

A function $f: I^2\rightarrow R$ is called Hölder continuous if $|f(\mathbf{x})-f(\mathbf{y})|\le||\mathbf{x}-\mathbf{y}||^\alpha$. However, what is the meaning of "f is $1/2$-Hölder continuous in ...
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Is $f(x)=\frac{\sin x}{x}$ for all $x\neq 0$ differentiable?

The function $f(x)$ is defined by $$f(x)=\frac{\sin x}{x}$$ for any $x≠0$. For $x=0$, $f(x)=1$. My work: Determine if the function is continuous, differentiable and if the latter, find its ...
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If $f$ is a real function and $g(x)=\lim_{t\to x}f(t)$, then $g$ is continuous?

Suppose that a real function $f$ has a limit at every point in a set $K\subset\mathbb{R}$ and $$g(x)=\lim_{t\to x}f(t)$$ Does that imply that $g$ is continuous on $K$?
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$X$ compact, limit of $f$ exists for all $p\in X$ then $f$ uniformly continuous on $X$ minus a countable subset of $X$.

For simplicity, let's take our compact set $X$ to be a subset of $\mathbb{R}$. Define $f:X\to\mathbb{R}$ to have a limit at every point in $X$. Then $f$ is uniformly continuous on a set $X$ minus a ...
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From continuity to uniform continuity

Consider a convex, open, non-empty set $B \subset \mathbb{R}^k$ and a function $V: B\rightarrow \mathbb{R}$ convex (and, hence, continuous) in $B$. Consider $\epsilon \in \mathbb{R}$, $\epsilon>0$. ...
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Paradox: Is the derivative of this function continuous at $x=0$?

\begin{equation} h(x)= \begin{cases} x^2 \sin(\frac{1}{x})&\text{ if } x\neq 0\\ 0&\text{ if } x=0 \end{cases} \end{equation} Is the derivative of $h(x)$ continuous at $x = 0$? How about the ...
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Continuity of a function - u(x,y) - Sequence is not converging by the function itself

Let the real function of two real variables$$u(x,y) = \begin{cases} x, & \quad \text{if } |y|>|x| \\ -x, & \quad \text{if } otherwise \\ \end{cases} $$ Is there a sequence $\{...
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47 views

Complex continuity proof

Show that continuity of complex functions holds under addition. We want to show given two functions $f,g\colon A\to\mathbb{C}$, where $A\subset\mathbb{C}$ is a domain, and some sequence $(a_n)$ ...
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54 views

Example of a function that is not continuous at $(0, 0)$ but continuous when restricted to any curve approaching the origin

I am looking for a continuous function $f: \mathbb{R}^2-\{0, 0\} \to \mathbb{R}$ satisfying the following condition: For any $g, h: \mathbb{R}\to\mathbb{R}$ continuous functions with $g(0)=h(0)=0$, we ...
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1answer
35 views

proof inequality . Thomae's function

I was looking at this post here, I'm trying to understand but I do not understand this: Let $x\in \mathbb{Q^+}$. How I can show $x > f(x)$. The function $f$ is defined by $f(x)=x$ if $x$ is ...
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Give an example of a equicontinuous that does not converge uniformly

Give an example of a equicontinuous sequence of functions ($f_n$) over a non-compact set $S\subset\Bbb R^n$ converging pointwise to a function $f$ at each $x\in S$, but $f_n$ does not converge ...
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35 views

Continuous limits

How can I show $f:[a,b] \rightarrow \mathbb{R}$ is increasing implies it is left continuous on $(a,b)$? My attempt: Case 1: $f(a)=f(b)$ Trivial case, it is a constant continuous function so it is ...
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122 views

Can someone provide some examples to illustrate the difference between Pointwise equicontinuity and Uniform equicontinuity?

I don't know what is with the subject of pointwise and uniform equicontinuity, pretty much all the material you can find online are either: Proofs i.e. pointwise equicontinuity is uniform ...
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Continuity of maps between metric spaces

I want to determine for the following maps $A\rightarrow B$, for which points of $A$ the map is continuous: $(\mathbb{C},d_E)\rightarrow(\mathbb{C},d_E)$, $z\mapsto \left\{ \begin{array}{...
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1answer
12 views

Proof that distinct numerator polynomials are equal for all x when over the same denominator polynomial

I am just curious about this part of the proof. The question is this: Suppose that F, G, and Q are polynomials and $\frac{F(x)}{Q(x)}=\frac{G(x)}{Q(x)}$ for all x except $Q(x)$ = 0 Prove that $...
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1answer
45 views

Find the joint probability density given the support set

Suppose that the support set of $(X,Y)$ is $$S_{X,Y}=\{(x,y)\in\mathbb{R}^2: x \geq 0 \text{ and } 0 \leq y \leq e^{-x/3}\}$$ $(X,Y)$ is uniformly distributed on $S_{X,Y}$. a) Find the joint ...