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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Example of continuous function over $\mathbb R^n$

Let $f:[0,1]\to\mathbb R^n$ such that $f(t)=ty+(1-t)x$ for all $x,y \in \mathbb R^n$. Prove that $f$ is continuous. I know a definition that A function $f\colon X \rightarrow Y$ between two ...
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How to tell if a function is continuous at (0,0)

I have to decide if the following function is continuous at (0,0). it's f(x,y) = x^2 + y^2 if (x,y) does not = 0, and f(x,y) = 2 if (x,y) = (0,0) so for the first one, I assume it is continuous, ...
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Continuous functions and open sets

I'm working on a proof and having trouble applying a certain theorem. I want to prove that if $ f $ is a continuous function from a metric space into the real numbers, then the set $ {f(x)>0} $ ...
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1answer
22 views

Proof: Cauchy sequences and uniform continuity

I'm working on a proof and I'm having trouble relating definitions I want to prove that if f is uniformly continuous, then if a sequence $ {a_n} $ is Cauchy, $ {f(a_n)} $ is Cauchy. So if $ f $ is ...
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0answers
24 views

Continuous and additive function is linear [duplicate]

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and $f(x+y)=f(x)+f(y)$, show that $f(x)=kx$, $k\in \mathbb{R}$. I tried to define $g(x)=f(x)-kx$ and $g(0)=0 $ but don't know how to ...
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30 views

Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
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Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty) $ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
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Riemann Integral Property for Continuous, Monotonic, Non-negative Function

If $f$ is continuous, non-negative, and monotonically increasing function on $[0,∞)$, then prove that $\int^{x}_{0} f(t)dt\leq xf(x)$ $\forall x ≥ 0$ My attempt: Define $F(x)=\int^{x}_{0} f(t)dt$. ...
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26 views

solve equation with Intermediate value theorem…

set $a_1$,$a_2$,$a_3>0$ and $λ_3>λ_2>λ_1$ on $ℝ$. show that there are exactly two $x$’s for $a_1/(x-λ_1) + a_2/(x-λ_2) + a_3/(x-λ_3) = 0$ I tried use the intermediate value theorem but I ...
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The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that f is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So ...
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three elementary problems on limits of several variable . [on hold]

I'm learning limits of several variable new. Can anyone help me? Computing the following limits: $\lim_{(x,y)\to(0,0)}|x|^y$ $\lim_{(x,y)\to(0,0)}\sin(x/y)$ $\lim_{(x,y)\to(0,0)}x^2\cdot ...
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2answers
31 views

Properties for functions $f:[a,b] \to \mathbb R$? [on hold]

Let $f:[a,b] \to \mathbb R$ be a function. Which of the followings are true: A) If $f(x)$ is continuous then it is bounded. B) If $f(x)$ is continuous then it is increasing. C) If ...
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2answers
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Questions about absolutely continuous function

It could be a silly question but the definition of absolutely continuous function says that "A real valued function $f:[a,b]\rightarrow\mathbb{R}$ is absolutely continuous on $[a,b]$ if for all ...
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4answers
42 views

Prove a Continuous Distribution Function is Uniformly Continuous

Let $F$ be the distribution function for a random variable $X$ and it is given that $F$ is continuous over the entire real line. Prove that $F$ is uniformly continuous over the real line. My ...
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derivative of a function has only 2nd kind discontinuities

How would I be able to show the following claim If $f$ is differentiable with a finite derivative in an interval, then at all points $f'(t)$ is either continuous or has a discontinuity of the second ...
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1answer
25 views

Smooth saturation function

I need a function similar to $$Saturation(x)=min(max(x, -1), 1)$$ except for I need it to be smooth with no jump in its derivatives. It seems $arctan$ is not a good candidate since I need it to keep ...
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1answer
19 views

Sequential continuity on metric spaces

Please give me a hint for proving this statement: Let $(X,d)$ and $(Y,d')$ be metric spaces, $f$ a function from $X$ to $Y$. If $f^{-1}(B) $ is closed in $X$ for all closed subset $B$ of $Y$, then ...
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2answers
31 views

A sequence of Continuous Functions Converges Uniformly over $\mathbb{R}$ if it Converges Uniformly over $\mathbb{Q}$

I'm trying to show that if ${f_n}$ is a sequence of real functions that is continuous over all of $\mathbb{R}$ and that converges uniformly to $f$ over $\mathbb{Q}$, then it converges uniformly to $f$ ...
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For every continuous function $f:[0,1]\to[0,1]$ there exists $y\in [0,1]$ such that $f(y)=y$ [duplicate]

I want to prove that if we have a continuous function from the closed interval [0,1] to the closed interval [0,1], that there exists a value y in [0,1] such that $f(y)=y$. I have an idea of a theorem ...
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example of a monotone non-continuous map.

Let me start by defining some terminology to be sure I made no errors there. Parts of this are translated freely from my mother tongue so feel free to correct terminology or the definitions themselves ...
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If $f$ is continuous and bounded on $(a,b)$, is it true that $f$ is Riemann-integrable on $[a,b]$? [on hold]

If $f$ is continuous and bounded on $(a ,b)$ or $(a, b]$ or $[a, b)$, is it true that $f$ is Riemann-integrable on $[a,b]$? Thanks.
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+50

Continuity and Differentiation example

I have proved that $g$ is continuous on $(0,2)$ and I just wish to check if my solution for $g$ being right continuous at $0$ and hence continuous at $0$ is correct. $\lim\limits_{x \to 0+}g(x) ...
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Continuity of function $f(x) = \lim_{n \rightarrow \infty } (cos(x-\pi/4))^n$

How to check continuity of such function? For me it will be continuous for $x \neq \pi/4+n\pi$ because for such x $|(cos(x-\pi/4))|<1$ But I'm not sure. Can someone tell me if I'm wrong or not? ...
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Which one is true?

Please give explanation against each option: Which of the following statements are true? a. Let $\psi$ be a non-negative and continuously differentiable function on $]0,\infty[$ such that ...
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1answer
31 views

Regarding sup and inf of a continuous function

Suppose $f:\mathbb R\to \mathbb R$ be a continuous function such that $\lim\limits_{x\to \infty}f(x)=0=\lim\limits_{x\to -\infty} f(x)$. Then I want to show that $f$ is bounded and attains at least ...
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1answer
40 views

Determine c and d so that f(x) is continuous if.. [closed]

$$f(x)=\begin{cases}2x^2+cx+d &\text{ for }x<-3\\4&\text{ for }x=-3\\dx^2+2x+c&\text{ for }x>-3\end{cases}$$
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0answers
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How to estimate norms involving $|a-b|$?

I know the title isn't the best one. Here's my problem: Whenever I'm given functionals such as: $$\phi: \mathcal{C}^1([0,1]) \ni f \rightarrow f(\frac{1}{3}) - f'(\frac{2}{3}) \in \mathbb{R}, \ \ ...
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Prove this function is lower semi-continuous

Let $X$ be a metric space, and $B$ his borel $\sigma$-algebra. Fix $r>0$ Let $\mu$ be a probability measure on $(X,B)$ and define $f(x)=\mu(B(x,r))$. Show that $f$ is lower semi continuous. What ...
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$l_2$ sequence, series with square root

I'm trying to prove that the following functional is continuous: $$\phi : \mathcal{l}_2 \ni \{x_n \} \rightarrow \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}x_{3n} - \sum_{n=1}^{\infty} \frac{1}{n}x_{2n} ...
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1answer
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Polynomial, bounded functional

In order to prove continuity of the functional $$\varphi: \mathbb{R}[X] \ni p \rightarrow p'(2011) \in \mathbb{R}$$ where $$||p|| = \sup \{|p(t), \ t\in [0,1]\}$$ I'd like to prove that this ...
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counter example of equicontinuous

Consider the functions on $[0,1]$: $f_n(x)=nx$, when $x$ is between $0$ and $1/n$ $f_n(x)=2-nx$, when $x$ is between $1/n$ and $2/n$ $f_n(x)=0$, otherwise How to see it is not (uniformly) ...
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Finding the limit of a sequence of sequences

Take any $\bar{r} \in \mathbb{R}$ with $\bar{r}>0$. Assume that $f : \mathbb{R} \rightarrow \mathbb{R} $ is continuous. Assume that for all $r\in [0,\bar{r})$, there exists a strictly decreasing ...
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1answer
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Find the maximum number of a continuous function

Lets define a function $z:\mathbb{R}^\mathbb{R}\to\mathcal P(\mathbb R)$ that gives you the set of zeros of any $\mathbb R ^\mathbb R$ function. Now, we define a set $S=\{z(f):f\in\mathbb R ^\mathbb ...
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1answer
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Analyze the continuity of the following function

Here in my book I have such an exercise with the explanation given below, but still there is something the authors didn't add, but simply put "...after some operations...". Here is such an exercise: ...
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Relations between $\varepsilon$ and $\delta$ in the $\varepsilon-\delta$-Defintion of Continuity

A function $f : \mathbb R \to \mathbb R$ is continuous at $x_0$ iff for each $\varepsilon > 0$ there exists some $\delta > 0$ such that $$ |x - x_0| < \delta \Rightarrow |f(x) - f(x_0)| < ...
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5answers
682 views

Issue with Spivak's Solution

Here was the problem: Here is the solution from his solutions book: This is barely a proof. How can he just say let $f(c) = 0$? How do you prove that $f(c) =0$ and how do you prove that $f(d) ...
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1answer
44 views

$f: \mathbb R \to \mathbb R$, define $f^2(x)=f(f(x))$

Given $f: \mathbb R \to \mathbb R$, define $f^2(x)=f(f(x))$, then which of the following statements are true: $1.$ $f$ is strictly monotonic then $f^2$ is strictly increasing. $2.$ ...
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1answer
38 views

Properties of Linear subspace of function space

The situation is the following: Suppose $X$ is a compact topological space. Let $A\subset C(X)$ be a linear subpace of the space of continuous functions on $X$. Assume that we know that for $x,y\in ...
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1answer
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Constructing a Continuous Everywhere but Nowhere Differentiable Function

In Neal Carothers' Real Analysis he claims that $$f(x)=\sum_{k \mathop = 0}^\infty 2^{-k}g(2^{k}x)$$ is a continuous but non-differentiable function over the real line if $g(x)$ is the distance ...
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Continuity of the function $f$ defined by $f(x,y)=1$ if $xy=0$ and $f(x,y)=2$ otherwise

Define $f:\mathbb R^{2}\to \mathbb R$ by $$f(x,y)=\begin{cases} 1 &\text{ if } xy=0\\2 &\text{ otherwise } \end{cases}$$ Let $S$ be the set of all those points of $\mathbb R^{2}$ at which ...
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Existence of continuous angle function $\theta:S^1\to\mathbb{R}$

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all ...
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1answer
36 views

uniformly continuous function is bounded

I saw already someone proved this question. for a bounded set $E$, if $f$ is uniformly continuous on $E$ then $f$ is bounded on $E$ proof what I saw is by dividing $E$ into $n$ ...
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2answers
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Does there exist a green's function that does not have translation symmetry?

I noticed that most Green's functions I have used take on the following functional form $G(x_1,x_2)=G(|x_1-x_2|)$. I assume these subsets of Green's functions are translationally invariant? Correct me ...
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101 views

Discontinuous Differentiable and One to One

If the derivative of a function (from $\mathbb{R} \rightarrow \mathbb{R}$) at a point $x_0$ is discontinuous, does that imply that the function is not one to one or injective in a neighborhood of ...
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Question on Uniform Continuity

Is it generally true that all uniformly continuous bijections $f: X \to Y$, where $X$ and $Y$ are metric spaces, have uniformly continuous inverses? If not, then what would be a counterexample, and is ...
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1answer
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Are there any other functions that are always discontinuous?

Looking at a graph (or finding some solutions with a calculator) of the function $f(x)=n^x$, where $n<0$, shows that it has one interesting property, so far as I can see: It is discontinuous ...
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0answers
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To prove existence of an open set of functions

I am trying to prove the following: In $C(X,Y)$ with $X=[0,1]$ and $Y$ of finite dimension $K$, $C(X,Y)$ having the topology of uniform convergence, for any $K$ finite there exists an open set of ...
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1answer
16 views

Bounded function with parameter

Let f be defined $f:(a,b) \rightarrow \mathbb{R}$, $f$ is continous and $a,b \in \mathbb{R}$. If $a=-\infty$ or $b=\infty$, is the function bounded? I do not know how to figure this out, detailes and ...
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continuity of the function with parameter

$f:(a,b)\rightarrow \mathbb{R}$ is continous $a,b\in \mathbb{R}$ Investigate the existance of a limit $\lim_{x\rightarrow a} f(x)$. Do not understand this utterly, please help.
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2answers
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Intermediate value theorem using [closed]

Consider $f:\left[0,1\right]\rightarrow \left[0,1\right]$, continuous function. For which values of $a$ there must be exist $c\:\in \left[0,1\right]$ such that $f\left(c\right)\:=\:a\cdot c$?