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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Prove $f:\mathbb{N} \rightarrow \mathbb{R}$ is continuous using the definition of sequential continuity

The definition of sequential continuity is that $x_n \rightarrow x \implies f(x_n) \rightarrow f(x)$. If the terms of the sequence $\{x_n\}$ are only natural numbers, I know that for all $\epsilon ...
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18 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
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2answers
66 views

Is this a criterion for continuity?

Given a topological space $(X,\tau)$ and the product space $(X^2,\tau_2)$. Define the diagonal $\Delta X^2=\{(x,x)\,|\,x\in X\}$ and a set $\mathbf S_\tau=\{\mathcal A\in\tau_2|\Delta ...
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3answers
51 views

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true? (a). If $g$ is continuous, then $f\circ g$ is continuous. ...
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Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
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2answers
45 views

Prove that a continuous inverse-transformation of $f: [0,1) \cup \{ 2 \} \to [0,1]$ exists

I am having this transformation $f: [0,1) \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ I've already proved that it is continuous. Question: Is ...
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2answers
29 views

Existence of continuous functions $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ ; and what if $(0,1)$ replaced by $[0,1)$ ?

Does there exist continuous functios $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ \ $g\big((0,1)\big)$ ? The problem I am having is that since $(0,1)$ is not compact I am not able to tell ...
3
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1answer
44 views

How to proove that a bijective transformation is NOT continous

I am having this transformation $f: \mathbb R \to \mathbb R$ $$f(x) = \begin{cases} x & x \in \mathbb R \setminus \mathbb Q \\x+1 & x \in \mathbb Q \end{cases}$$ I've already prooved ...
3
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46 views

Teacher Challenge - multiple parts

This was his challenge: "I would like you to consider the function $x^{r+\alpha}$, $r$ is an integer, $\alpha$ is a real number between $0$ and $1$. Differentiate it until you get a singularity ...
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15 views

A basic doubt on upper semi-continuity of set-valued maps

Upper Semi-Continuity for set valued maps have two definitions $h:\Bbb R^d \to 2^{\Bbb R^d}$ is upper semi-continuous if 1) Sequential definition : $x_n \to x$, $y_n \to y$ and $y_n \in h(x_n)$ ...
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0answers
13 views

Requirements for existence Lebesgue-Stieltjes measure corresponding to distribution function in $\mathbb{R}^n$

I am going through Ash's book "Probability and Measure Theory". It says that: We know that a distribution function of $\mathbb{R}$ determines a corresponding Lebesgue-Stieltjes measure. This is true ...
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4answers
103 views

Prove that $f\left(x\right)=\sin\left(x\right)$ is Continuous.

The function $f\left(x\right)=\sin\left(x\right)$ is obviously continuous. But how would you prove this using the $\delta,\varepsilon$ definition of continuity? So given $x\in\mathbb{R}$ and ...
3
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2answers
75 views

Is $f: [0,1[ \cup \{ 2 \} \to [0,1]$ continuous?

I am having this transformation $f: [0,1[ \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ How can I prove that this transformation is continuous or ...
3
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27 views

Showing points of continuity of a function f(x) that takes the value 1/n whenever x belong to a sequence {An} and is zero elsewhere.

I am given a sequence $(An), n=1,2,3,...$ which consists of distinct numbers, which converges to $3$ as $n$ tends to infinity, but none of its terms are equal to $3$. Then I am given a function $f(x) ...
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1answer
35 views

Is function $f$ also uniformly continuous?

I've been thinking on the following problem lately: Let $(X,d)$ be a metric space and $f_1,f_2,...,f_n: X \rightarrow \mathbb{R}$ and $f(x) = \max\{f_1(x),f_2(x),...,f_n(x) \}$,$x\in X$ If the ...
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$X\subset \mathbb R$ and $f,g:X\to \mathbb R$ be continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)=X$ [on hold]

Let $X\subset \mathbb R$ and $f,g:X\to \mathbb R$ be continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)=X$, Which of the following sets can not be equal to $X? $ A. ...
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34 views

Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...
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0answers
33 views

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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3answers
47 views

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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1answer
36 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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3answers
63 views

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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43 views

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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1answer
25 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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3answers
51 views

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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4answers
40 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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3answers
50 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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3answers
38 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 ...
5
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2answers
139 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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0answers
35 views

Continuity of normal curvature

I want to show that the normal curvature is a continuous function. At first, here is the definition of normal curvature at point $p \in M \subset \mathbb{R}^3$ in direction $\mathbf{u} \in T_{p}M$: ...
3
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1answer
54 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
34 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
66 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 $f$ ...
0
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1answer
22 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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35 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 ...
1
vote
3answers
96 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
54 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
39 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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1answer
38 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
49 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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34 views

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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2answers
74 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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48 views

Prove a sequence converges to f(A).

I would like to know if this is an accurate proof
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65 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
23 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
35 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
21 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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2answers
24 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 ...
1
vote
1answer
39 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. ...
2
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0answers
42 views

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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votes
1answer
22 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) ...