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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Proving uniform continuity and uniform discontinuity

Could someone please explain to me how to show uniform continuity and not uniformly continuous for the following: $f(x) = \frac{1}{x^2}$ for $A = [1, \infty)$ show uniform continuity $f(x) = ...
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Need to show the following function is uniformly continuous on R

Could you please tell me how I am supposed to show that $f(x) = \dfrac{1}{(1+x^2)}$ is uniformly continuous in $\mathbb{R}$. I did some pre-calculation and found that $|f(x) - f(u)| < \epsilon$ if ...
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38 views

Show that $f(z):=\sum a_n (z-z_0)^n$ is continuous whenever $z$ is in disk of convergence.

Consider a power series $\sum a_n(z-z_0)^n$, and assume it has radius of convergence $r$. Then we know that $\forall z\in(z_0 -r,z_0 +r)$, this power series converges absolutely by root test. Thus we ...
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How to show that a real continous function with image in the rationals is constant?

Can someone please explain to me how I am supposed to approach this question: If $f: [0,1] \to \mathbb{ R}$ is continuous, and has only rational values, then $f$ must be a constant.
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How to approach this problem?

Could someone please explain to me how I am supposed to approach and prove the following problem: Let $I= [a,b]$ and $f:I \to \mathbb{R}$ be a continuous function on $I$ such that for each $x \in I$, ...
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30 views

If a continuous function is positive on a closed interval $I$, there exists a positive number $\alpha$ such that $f(x) > \alpha$ for all $x\in I$

Could someone please explain to me how I am supposed to how I am supposed to approach this question: Let $I = [a,b]$ and $f:I\mapsto \Bbb R$ be a continuous function on $I$ such that $\forall x\in ...
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Continuity set of a difference of two upper semi-continuous real functions over a metric space [closed]

The difference of two upper semi-continuous functions is in general neither upper- nor lower semi-continuous. But what can be said about the continuity set of such a functions, specifically its ...
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Defining Topological Continuity

I have seen this definition many times: Topological Continuity: A function $f:X\rightarrow Y$ is continuous if for all open sets $U \subseteq Y$, the preimage $f^{-1}(U)$ is open in $X$. I don't ...
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48 views

Need help in the continuity question [duplicate]

could someone please explain to me the following question: Let $f,g$ be continuous functions from $\mathbb{R}$ to $\mathbb{R}$ and suppose that $f(r) = g(r)$ for all $r \in \mathbb{Q}$. Is it true ...
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53 views

Find two functions $f$ and $g$ such that they are both discontinuous at $c$, however, $f+g$ and $f\cdot g$ are both continuous at $c$

Could someone please explain to me how to approach these kinds of question and also what is the answer to the following question? Give an example of a function $f$ and $g$ such that they are both ...
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39 views

No continuous transformation $f([a,b])= ]a,b[$

$ a,b\in\mathbb R$ with $a<b $. Now I want to show that there is NO continuous transformation $f: [a,b] \to \mathbb R $ with $f([a,b])= ]a,b[$ How can I proove that this transformation don't ...
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147 views

$f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?

Let $f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$? A. $\{0\}$. B. $(0,1)$. C.$[0,1)$. D.$[0,1]$. ...
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1answer
61 views

Theorem 4.22 from baby Rudin: continuity and connectedness

I have some parts that I don't understand from the given proof. The theorem is: If $f$ is a continuous mapping of a metric space $X$ in to a metric space $Y$, and if $E$ is a connected subset of $X$, ...
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1answer
40 views

Extreme point of unit balls, over $\mathbb C$

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $\mathcal{C}[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
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2answers
69 views

The function $\frac1x$ is an homeomorphism

I have the function $f:(0,+\infty)\rightarrow (0,+\infty)$ defined by $f(x)=\frac1x$ I want to prove that $f$ is an homeomorphism. So I have that $f$ is surjective or onto by definition and that $f$ ...
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1answer
24 views

On the continuity of $xf(x)$ and $x^2f(x)$, where $f$ is the Dirichlet function

Let $$f(x) = \begin{cases}1\qquad x\in\mathbb{Q}\\ 0\qquad x\notin\mathbb{Q} \end{cases}$$ Then how do I show that $xf(x)$ is continuous in $0$ and that $x^2f(x)$ is differentiable there as well? ...
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36 views

Extreme point of unit balls, the complex case [duplicate]

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $C[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
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3answers
98 views

Continuous functions and infinum

Let $f:\mathbb R \to \mathbb R$ with $f(-2)=4$ and $f(3)=7$. Let $S:=\{x \in [-2,3]\mid f(x)\geq 5\}$. Then $\alpha:=\inf S$ exists. If $f$ is continuous at $\alpha$, show that: (a) ...
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1answer
57 views

Using continuity to prove f is a constant function

Recently missed this problem on an exam. Just went to office hours to clarify what the proper proof was and wanted to see if, in attempting to repeat the problem, I can figure out if there are better ...
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1answer
44 views

Brownian motion is almost surely continuous

Why is Brownian motion required to be almost surely continuous instead of merely continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
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40 views

Iterating average

If $f$ is a continuous function $[0,1]\to \mathbb R$, we define a linear application $T$ as follows $$T(f)(x)=\begin{cases} f(0) & \mathrm{if }~ x=0 \\[0.2cm] \displaystyle ...
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1answer
54 views

On consequences of $\int_{0}^1f(x)x^ndx=0 , \forall n \in \mathbb Z^+\cup\{0\}$

If $f : [0,1] \to \mathbb R$ is a continuous function and $\int_{0}^1f(x)x^ndx=0 , \forall n \in \mathbb Z^+\cup\{0\}$ then is it true that i) $\int_{0}^1(f(x))^2dx=0$ ? ii) ...
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Understanding the definition of continuity from real analysis

I've stared at and worked with the definition of continuity of a real valued function at a point for many (like $3$) years, but there are some things that have always bothered me about it. First, ...
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76 views

Show that a function is constant

Let $S$ be a non-empty set of real numbers such that if $a,b$ are distinct elements in $S$, then $|a-b|\geq 1/2014$. Let $f:\mathbb R \to \mathbb R$ be such that the range of $f$ is a subset of $S$. ...
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50 views

Prove a sequence converges to f(A).

I would like to know if this is an accurate proof
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70 views

For a $C^1$ function, the difference $|{g'(c)} - {{g(d)-g(c)} \over {d-c}} |$ is small when $|d-c|$ is small

Suppose $g\in C^1 [a,b]$. Prove that for all $\epsilon > 0$, there is $\delta > 0$ such that $|{g'(c)} - {{g(d)-g(c)} \over {d-c}} |{< \epsilon }$ for all points $c,d \in [a,b]$ with $0 ...
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1answer
26 views

Floor function and continuity

In the topic Proof concerning definite integral, I've received down-votes because I said that the function $f(x) = \lfloor x \rfloor$ is continuous in $[a, b]$, for $0 < a < b<1$. Why am I ...
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1answer
38 views

Two statements about one-sided derivative and monotony

The statement 1 is: $f\colon [a,b]\to\mathbb R$,continuous on $[a,b]$,$f'_-(x)$ exists and is $\le0$ for all $(a,b]$.Can we infer that f is non-increasing on $[a,b]$? My attempt is: Assume $f$ is not ...
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1answer
22 views

Derivative of a function containing indicator function?

Consider $\delta\in \mathbb{R}$ and $X \in \mathbb{R}$. Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a map defined as $$ f(X; \delta):=\delta*1\{X\geq 0\}+X $$ where $1\{X\geq 0\}$ is $1$ if $X \geq ...
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30 views

Continuous function positive at a point is positive in a neighborhood of that point

Pretty much the problem asks if a function is continuous at the point $c$ and $f(c) > 0$ then there exists a $d > 0$ such that $\forall x$, $f(x) > 0$ with $|x-c| < d$. I can understand ...
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1answer
45 views

What non-integer number has the smallest factorial? [duplicate]

Quick google search for factorial of non-integers led me to gamma function. I tried that in my calculator and it worked as expected for non-integers. Perhaps implements gamma function internally. ...
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Is $f$ continuous if for every $p$, there is a sequence $p_n \to p$ such that $f(p_n) \to f(p)$?

Let $(X, d)$ be a metric space and $f : X \rightarrow X$ a function that satisfies the following property: For every $p \in X$ there exists a sequence $\{p_n\}\subset X$ such that $p_n \rightarrow p ...
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1answer
28 views

Continiuous functions to the sphere

Let $X=AUB$ be a topological space and $A, B$ be a two closed set of X. Let $f:A\to S^n$ and $g:B\to S^m$ be two continuous functions. Define $h:X\to S^{n+m+1}$ by $$h(x)=(f(x),0,\cdots , 0) ...
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For every periodic continuous function $f$, the function $s\to \int_a^b f(x/s)\, dx $ is continuous

Let $f: \mathbb R \to \mathbb R$ be a continuous function such that $f(x+1)=f(x)$ for all $x\in \mathbb R$. Fix $a$ and $b$ such that $a<b$, and define a function $g: \mathbb R \to \mathbb R$ by ...
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53 views

Prove functions are continuous

$X$ and $Y$ are metric spaces and $f$ is a function from $x$ to $y$. Prove that $f$ is continuous at $p$ if and only if f maps all sequences that converge to p to all sequences that converge to f(p) ...
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1answer
37 views

Show that $|sin(x)+cos(x)|$ is continuous at $\pi$

Show that the function $f(x)= |\sin(x)+\cos(x)|$ is continuous at $x=\pi$. By drawing the graph, we can easily show that it is continuous, but how can we show it by using limits. Please help.
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1answer
31 views

A continuous integer-valued function on a compact metric space has finite range

Let $X$ be a compact metric space and let $f:X\to\mathbb Z$ be a continuous function. (Here $\mathbb Z$ has the Euclidean topology induced from $\mathbb R$.) Prove that $f$ can assume only finitely ...
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45 views

Continuity in the complex plane

I was reading a book where it is claimed that a sufficient condition for \begin{equation} f(x)=\frac{1}{2\pi}\left|\sum_{j=0}^{\infty}\theta_je^{ix j}\right|^2 \end{equation} to be continuous and is ...
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Is a continuous bijection function from a hausdorff space to a compact space is a homeomorphism?

We know a continuous bijection from a compact space to a Hausdorff space is always a homeomorphism. But I am wondering what happened if we switch the domain and codomain. Is a continuous bijection ...
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Function continuous at irrationals and discontinuous at rationals [duplicate]

Q: Given the function $f(x)=\sum_{n=1}^\infty f_n(x)$, where $f_n(x)= \left\{ \begin{array}{lr} 0; \;\;if \;x< r_n \\ \displaystyle \frac{1}{2^n}; x\geq r_n ...
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1answer
57 views

If a limit does not exist does that make it unequal to some given value?

I was asked to pick a function $f$ for which $\lim_{x\to c^-} f(x) \neq \lim_{x\to c^+} f(x)$ for some $c$. I used $f(x)=\sqrt{x-2}$ with $c=2$ as an example of such a function. My question is the ...
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1answer
39 views

Show that $f(a)=f(a+\frac{1}{n})$ for some $a \in [0, 1-\frac{1}{n}]$, given that:

Show that $f(a)=f(a+\frac{1}{n})$ for some $a \in [0, 1-\frac{1}{n}]$, given that: $f$ is continuous on $[0,1]$ and $f(0)=f(1)$. $f(a)=f(a+\frac{1}{2})$ for some $a \in [0, 1/2]$. $n\in \Bbb N$ and ...
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2answers
54 views

Does this piecewise function contradict the fact that all differentiable functions are continuous?

I learned that all differentiable functions are continuous. Why does the following equation not violate this rule: $$f(x)=\begin{cases}x^2+3 \quad &\text{when } x>1 \\ x^2 \quad ...
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1answer
65 views

How to use Cauchy-Scharwz inequality to prove differentiable?

I'm attempting to understand how to prove the function f such that $$f(x,y)=\frac{x^3y}{x^4+y^2}\;if\;(x,y)\neq (0,0)$$ $$f(x,y)=(0,0)\;if\;(x,y)=(0,0)$$ is continuous in $\mathbb R^2$. The solution ...
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1answer
61 views

About the continuity of $f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k}$

Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$. Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} ...
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34 views

Proposed proof of continuous operator on Sobolev space

Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow ...
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2answers
76 views

Prove that $f $ is constant

Let $f:\mathbb R \to \mathbb R $ be a continuous function such that for all $x \in \mathbb R$, $f(x)=f(x^2) $ prove that $f$ is constant. "please give me hints not answer. thanks a lot. :):):):):)" ...
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1answer
27 views

Finding continuous functions from a set

Let $A=\{0,1,\frac{1}{2},\frac{1}{3},...\}$. I want to find continuous functions from $f:A\to \mathbb R$. I proceed in this way. Any sequence converges to $x(\neq 0)$ will be eventually constant ...
3
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3answers
78 views

prove $f(x)=x$ has a unique solution

Question: Let $f$ be a continuous function from $\mathbb{R^2} \rightarrow \mathbb{R^2}$ such that $| f (x)− f (y)| ≤ \frac {1}{3} |x−y|$. Prove $f(x)=x$ has a unique solution. My sketch: There ...
2
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0answers
39 views

Shannon Entropy Continuity Constraint

I have the following problem: I want to find the probability density $p$ which maximizes the Shannon entropy \begin{equation} S := - \int_{x_b}^{x_c} dx ~ p(x) \log (p(x)) \end{equation} under the ...