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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246 views

Compactness implies Continuity?

I am stuck on this question (probably there are many counterexamples, but I can't find any). "Suppose $f:\mathbb{R}\mapsto\mathbb{R}$ that preserves compactness (i.e, for every $K \subseteq R$, then ...
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38 views

Extending an holomorphic function

Let $D \subset \mathbb{C}$ be a disc. Is there a function $$ f \in H(\mathop D\limits^{\circ} ) \cap C(\overline{D}) $$ such that , for every open set $A \supset \overline{D} \ $, $f \notin H(A) \ \ ...
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1answer
123 views

Is $\cos\sqrt{xy}$ uniformly continuous?

I'm trying to find out if $$ f(x,y)=\cos\left(\sqrt{xy}\right) $$ is uniformly continuous on the set $\{(x, y)\in\mathbb{R}^2 : x\geq0, y\geq 0\}$. The theorems I have available to use for this are ...
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86 views

Two-variable limit, quotient of polynomials

I'm trying to evaluate the following limit, $$ \lim_{(x,y)\to(0,0)} \frac{x^3-y^2}{x^2-y} $$ which I think it doesn't exist, since for the curve $\alpha :[0,1]\to \mathbb R^2$, $\alpha(t) = (t, t^2)$ ...
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1answer
42 views

Continuosly differentation on composite functions

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ a $C^1$ function and defined $g(x) = f(\|x\|)$. Prove $g$ is $C^1$ on $\mathbb{R}^n\setminus\{0\}$. Give an example of $f$ such that $g$ is $C^1$ at the origin ...
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0answers
63 views

show a function is continuous at irrational point

Suppose $\mathbb{Q} \cap [0,1]=\{r_1,r_2,r_3,...\}$ (We can do this because $\mathbb{Q}$ is countable and $\mathbb{Q} \cap [0,1] \subset \mathbb{Q}$. Let $x \in (0,1)$. Define ...
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231 views

On continuity of roots of a polynomial depending on a real parameter

Problem Suppose $f^{(t)}(z)=a_0^{(t)}+\dotsb+a_{n-1}^{(t)}z^{n-1}+z^n\in\mathbb C[z]$ for all $t\in\mathbb R$, where $a_0^{(t)},\dotsc,a_{n-1}^{(t)}\colon\mathbb R\to\mathbb C$ are continuous on ...
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1answer
86 views

calculus continuity of a hard question?

How do i calculate the continuity of a function if the functions are in a given set limit. I tried doing it but I epically failed… Please help!how do I solve it? When I tried to solve this I got that ...
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29 views

Krylov-Bogoliubov theorem without continuity

This question is very closely related to: Continuity in the Krylov-Bogoliubov theorem. The standard counterexample, which is presented in Katok-Hasselblatt is the following: Let $f:[0,1]\rightarrow ...
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1answer
54 views

Continuous injection from $\mathbb{R}^2$ to $\mathbb{R}^2$.

Could someone give me an example of a continuous injection from $\mathbb{R}^2$ to $\mathbb{R}^2$ which does not have a continuous inverse.
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1answer
28 views

continuity with lipschitz dertivative

Can somone explain me what does it mean to say a "function is continously differentiable with a Lipschitz derivative near the limit point". I dont understand the technical jargoans. Thanks
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26 views

limit of limit superior w.r.t truncated set

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
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3answers
44 views

Proving a fact about continuous function

Prove that if $f(a)>0$ and $f$ is continuous, then there is a $\delta >0$ such that for all $x$, $|x-a|< \delta$ implies $f(x)>0$.
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2answers
41 views

Is the function continuous at the indicated point?

$f(x) = x[x]$ at $x=2$ ($[x]$ is the greatest integer function) I am little confused, it seems like the function does exist at the given point. when limit goes to 2, $2[2] = 4$ and $f(2) = 4$ so ...
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1answer
24 views

If the function $f(x) =\lfloor \frac{(x-2)^3}{a}\rfloor \sin(x-2) +a\cos(x-2), \lfloor . \rfloor$ denotes …

Problem : If the function $$f(x) =\left\lfloor \frac{(x-2)^3}{a} \right\rfloor \sin(x-2) + a \cos(x-2),$$ where $\lfloor . \rfloor$ denotes the greatest integer function is continuous and ...
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1answer
24 views

How to show $F(\vec{x})$ is lipschitz on $[0,1] \times [0,1]$

We know that $f(x)=x^2$ is lipschitz on [0,1]; intuitively the steepest the graph of $f$ gets is at $x=1$, and I can find the lipschitz constant by a simple factoring argument. Now say ...
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476 views

A function takes every function value twice - proof it is not continuous

I want to prove the following nice statement I've found: A function $f: [0,1] \rightarrow \mathbb{R}$ takes every function value twice - proof it is not continuous I've already found an answer to my ...
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1answer
37 views

Showing a function is limited

This is something about which I'm quite confident, but I would really like to be sure Let $f(x)$ be a continuos function in $\mathbb{R}$ Then if $\displaystyle \lim_{x \to \pm\infty} f(x) = l \in ...
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1answer
73 views

prove $\exists$ a continuous function g s.t. $|g(x)-f(x)|<\epsilon$ for all x in a subset E of (a,b) and $\mu[(a,b)-E]<\delta$

I am having trouble starting with ths problem. Let f be a Lebesgue-integrable function over a bounded interval (a,b). Prove that for any $\epsilon >0$, $\delta >0$, there exists a continuous ...
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1answer
69 views

Problem of continuous real valued function

Which ofthe following statements are true? a.If $f:\mathbb R\to\mathbb R$ is injective and continuous, then it is strictly monotonic. b.If $f\in C[0,2]$ is such that $f(0)=f(2)$,then there ...
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365 views

Nonpiecewise Function Defined at a Point but Not Continuous There

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...
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1answer
49 views

Transformation of a continious function

Suppose that $f:[0,2\pi$] $\rightarrow \mathbb{R}$ is continuous and $f(0)=f(2\pi)$. Show that there exists an $x\in[0,\pi$] such that $f(x)=f(x+\pi)$. I simply have no idea where to start, any help ...
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1answer
45 views

Lower semi-continuity of a convex functional on $L^1(\Omega,[0,1])$

Let $\Omega$ be a bounded domain and $f:\Omega\times[0,1]\to[0,\infty]$ be such that $x\mapsto f(x,u)$ is measurable for every $u$, $u\mapsto f(x,u)$ is continuous and convex for a.e. $x$. Furthermore ...
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3answers
89 views

Prove that there is an $\varepsilon$ such that $f(x) > x + \varepsilon$ for a continuous $f(x) > x$ at $[0,1]$

I know this question was answered by using another theorem here but I wish I could get comments on my way of trying to prove it. We were asked to prove that for a function $f(x) > x $ which is ...
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2answers
40 views

Check whether the function is continuous at 0 - what went wrong?

I have to check the whether the following function is continuous: $$ \ f:\mathbb{R}\rightarrow \mathbb{R},~f(x)=\left\{ \begin{array}{lll} e^{1/x} &\text{if} & x < 0, ...
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2answers
81 views

Is this $\epsilon-\delta-$proof correct?

I have to Show that $$\mathbb{C} \rightarrow\mathbb{R}; z \rightarrow \Re z$$ is a continuous function using the $\epsilon-\delta-$criteria. So what I did is the following: I have to Show that ...
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1answer
56 views

Equivalence relation between measures $\nu$, $\mu$ is equivalent to $\nu = f \mu$ for a density $f$.

I'm working on an exercise that wants me to show that for $\sigma$-finite measures $\nu$ and $\mu$ the relation $\nu \sim \mu$ (defined by $\nu \ll \mu$ and $\mu \ll \nu$) is equivalent to $\nu = f ...
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1answer
68 views

Find a uniformly continuous function such that $a_{n+1}=f(a_n)$

$a_{n+1} = a_n - a_n^2$, $a_1 = 2/3$. for $n\ge1$ a) Show the series converges and find its limit. b) find a uniformly continuous $f:\mathbb{R}\rightarrow \mathbb{R}$ such that: ...
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2answers
73 views

How to solve for continuous functions? [closed]

$f: D \to \mathbb{R}$ is a function with $D \subseteq \mathbb{R}$ 1.The function is continuous in $a$ $\in$ $D$ if: $(\forall\epsilon>0)(\exists g>0)(\forall x \in D) [|x-a|< g \to ...
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0answers
57 views

Prove a function is uniformly continuous (might be Lipschitz Condition)

Let $f$, a continuous function defined on the interval $[0,\infty)$. It is given there are $a,b$ such that: $$\mathop {\lim }\limits_{x \to \infty } \left[ {f(x) - (ax + b)} \right] = 0$$ ...
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1answer
31 views

Prove that $\exists x_1,x_2 \in \mathbb{R}$ with $|x_1-x_2|=\frac{T}{2}$ such that $f(x_1)=f(x_2)$

Let $f:\mathbb{R} \to \mathbb{R}$ be continuous periodic function with T>0. Prove that $\exists x_1,x_2 \in \mathbb{R}$ with $|x_1-x_2|=\frac{T}{2}$ such that $f(x_1)=f(x_2)$. I don't even know ...
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1answer
343 views

Equicontinuity implies uniform convergence

So I know it's a theorem that if $\{f_n\}$ is a sequence in an equicontinuous family of functions defined on a compact metric space $K$ then if for all $x$, $f_n(x)\rightarrow f(x)$ pointwise then ...
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1answer
126 views

Prove that the function is uniformly continuous

Let $f(x)$ be a continuous function in $[0,\infty)$ there are $a,b \in \mathbb{R}$ such that $\lim_{x\to\infty} [f(x) - (ax +b)] =0$ prove that $f(x)$ is uniformly continuous in ...
3
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1answer
75 views

Existence of a continuous function.

Question is to check Which of the following statements are true? There exists a continuous function $f: \{(x,y)\in \mathbb{R}^2 : 2x^2+3y^2=1\}\rightarrow \mathbb{R}$ which is one-one. There exists ...
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1answer
54 views

Application of Mean Value Theorem: $f'(b) = \lim_{x\downarrow b} f'(x) = \lim_{x\uparrow b} f'(x) =: \gamma $

Let $a,b,c \in \mathbb{R}$ and $a<b<c$. Furthermore let $f: [a,c] \rightarrow \mathbb{R}$ be continuous let $f_{|(a,b)}$ and $f_{|(b,c)}$ be differentiable with $$\lim_{x\downarrow b} f'(x) = ...
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3answers
57 views

Finding examples of continuous functions

I'm looking for 1) a function that is discontinuous at 0, 1, 1/2, 1/3, 1/4, 1/5, ... but continuous everywhere else 2) a function that is discontinuous at 1, 1/2, 1/3, /4, 1/5, ... but continuous ...
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0answers
63 views

Space of Continuous mappings to metric spaces

I want to ask whether some basic result from the space $C([0,1],R)$, where $R$ is the real space carries over to the space $C([0,1],E)$, where $(E,\|\cdot\|_E)$ is a metric space. We know that ...
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28 views

Find an appropiate $\delta$ for continuty at a given point

Let $f(x)=1/x$. Show continuity at $x=1/2$ My work: $$\left| {x - \frac{1}{2}} \right| < \delta \Rightarrow \left| {\frac{1}{x} - 2} \right| < \varepsilon $$ $$\left| {\frac{1}{x} - 2} ...
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50 views

Is $f$ uniformly continuous

Prove/Disprove : Let $f:(0,1)→\mathbb{R}$ be Continuous. The condition $f(1/n)$$\rightarrow$$1/2$ and $f(1/n^2)$$\rightarrow$$1/4$ imply that $f $ is uniformly continuous.
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66 views

Fixed point and period of continuous function

Prove/ Disprove: Let $f:(0,1)\to(0,1)$ be such that $|f(x)-f(y)|\leq 0.5|x-y|$ for all $x ,y.$ Then f has a fixed point. 2.Let $f:\mathbb R\to\mathbb R$ be continuous and periodic with period ...
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1answer
67 views

Prove $mx+b$ is continuous at any point in $\mathbb{R}$

I need to prove, $mx+b$ is continuous at any point in $\mathbb{R}$ Now, as I have thought of there's 2 possible cases: 1) $m = 0$ 2) $m \neq 0$ So for case #2, $m < 0 \vee m > 0$ , and we ...
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1answer
78 views

Proving discontinuity

Assume set $A$ is countable and let$$f(x)=\cases{1 \text{ if }x\in A\\0\text{ if }x\notin A }.$$ Prove that $f$ is not continuous at $c\in A$. I've seen such a problem before where ...
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2answers
47 views

how to prove that the limit of this sequence of functions is continuous?

I have a norm that works in function space and that is $‖∙‖_{sup}:C([0,1])→R$, $‖∙‖_{sup}:=sup${$|f(t)|$}. I need to show that the metric is complete. So I need to show that every Cauchy sequence of ...
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1answer
15 views

Continuity and increasing functions

Suppose $f : D \to \mathbb{R} $ is continuous at the point $x_0 \in D$. Prove that: 1) if $f(x_0) > 0 $ then $\exists \delta > 0$ s.t. $|x-x_0| < \delta $ & $x \in D \implies f(x) ...
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1answer
57 views

continuous of a function

Let $U$ be a non-empty open set in $R^2$ and let $f:U\to R$ be a function. Suppose that the first partial derivactives of $f,f_1,f_2$ are defined and bouned on all of $U$. Show that $f$ is continuous ...
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1answer
62 views

Semicontinuous and monotone = Semicontinuous?

Is it true for a (upper/lower) semicontinuous function $f$ and a monotonically increasing function $g$ which is also (upper/lower) semicontinuous) that $f\circ g$ is also (upper/lower)semicontinuous? ...
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1answer
124 views

$f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous

Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
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2answers
24 views

Continuity of 'vectorial' function $\frac{x^2}{y^2-1}$

given is $f(x,y) = ( \frac{y}{x^2+1}, \frac{x^2}{y^2-1} ) $. I have to study the continuity of the function for$ (x,y) \to (0,1)$. First function $f_1$ is continuous, since $lim f_1 = 1/1 = 1$ so the ...
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1answer
40 views

Question regarding Continuity of F(x,y)

Let $f(x,y) = \begin{cases} \frac{2(x^3+y^3)}{x^2+2y}&\text{ } (x,y)\not=(0,0)\\ 0 &\text{ }(x,y) =(0,0). \end{cases}$ show that first order partial derivatives of $f$ wrt x and y exist at ...
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1answer
72 views

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 ...