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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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
107 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
83 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
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
58 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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45 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
80 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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2answers
55 views

Prove a sequence converges to f(A).

I would like to know if this is an accurate proof
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76 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
30 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
57 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
31 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
58 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
63 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
30 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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38 views

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 ...
0
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2answers
58 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
45 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
53 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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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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133 views

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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26 views

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
63 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
43 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
63 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
110 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
93 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} ...
3
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0answers
40 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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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
30 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 ...
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3answers
115 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
54 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 ...
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1answer
73 views

The set of all fixed points of a continuous function $f:[0,1] \to [0,1]$ , satisfying $f \circ f=f$ , is a non-empty interval?

Let $f:[0,1] \to [0,1]$ be a continuous function such that $f \circ f=f$ on $[0,1]$ , then is it true that the set $\{x \in [0,1] : f(x)=x \}$ is a non-empty interval? I can show that it is ...
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39 views

Prove that $f(x)$ is bounded. Please check my proof.

Assume $f:(0, \infty) \rightarrow \mathbb{R}$ is continuous. Also assume $\lim_{x \rightarrow 0}f(x)$ and $\lim_{x \rightarrow \infty} f(x)$ exist and are finite. Prove that $f(x)$ is bounded. ...
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3answers
149 views

Continuity is required for differentiability?

My professor emphasized that: Differentiability implies continuity and Continuity is required for differentiability. Since a function like $\frac 1 x$ is differentiable but not continuous, I ...
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1answer
64 views

Let $f(x)=(x+1)^2$. Prove that f is continuous at 0

I've started work from the definition, so for all $ϵ>0$, there is $δ>0$ such that $0<|x|<δ$, then $|(x+1)^2-(0+1)^2|<ϵ$. Then by expanding, $|x^2+2x|<ϵ$, $|x||x+2|<ϵ$, and by ...
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45 views

uniform convergence of sequence of function

I have a sequence of function $f_n$: $$ f_n(x) = \sqrt{x^2 + \frac1n} \qquad \text{on the interval } [-1,1] $$ and $$f(x) = |x| $$ I need to prove that the sequence of functions $f_n$ is uniformly ...
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21 views

Showing a map is a bounded linear operator.

Show that the map A : (C[0,1],∥·∥∞) → R, Ax = x(0), ∀x ∈ C[0,1] is a bounded linear operator. I know one has to show the map is continuous but I'm not sure how to go about proving it in this case. ...
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30 views

continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following: Does a continuous and concave function \begin{eqnarray*} f: N_{\mathbb{Q}} \to \mathbb{R} ...
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1answer
41 views

Prove that this integral is diverge

Let $f:[0,\infty) \to \mathbb R$ be a strictly decreasing continuous function, such that $\lim_{x \to \infty}f(x)=0$ prove that $\int_{0}^{\infty}\frac{f(x)-f(x+1)}{f(x)}$ is diverge.
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1answer
29 views

proof of continuous function for any real x

I have a function : $$ \sum_{n=1}^\infty \frac{sin(nx)}{n^2} \cdot x^2 $$ How is this function a continuous function for any $x \in \mathbb R $? I cannot prove it..
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25 views

uniform convergence of a function (continuous or differentiable or both?)

I have a function $S$: $$ S(x) = \sum_{n=1}^\infty \frac1{x+n^2} \\ \text{for} \ x \ge 0 $$ I need to determine if $S$ is continuous or differentiable or both.
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56 views

Proving continuity of a function at a point - Homework

$\Bbb R^2$ is using the Euclidean metric, $\Bbb R$ is using the standard $|y-x|$ metric. We define $f:\Bbb R^2\rightarrow\Bbb R$ by $$f(x,y) = \left\{\begin{array}{ll} \frac{x^6+y^6}{x^2+y^2} & ...
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1answer
45 views

Construct a sequence of continuous functions which converges pointwise to $\lfloor x \rfloor$

Suppose $f(x)=\lfloor x \rfloor$ for $x \geq 0$. Define a sequence of functions $(f_n(x))_{n \geq 1}$ where $f_n(x) = \left\{ \begin{array}{lr} x^n & : x \in [0,1)\\ (x-1)^n+1 ...
3
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2answers
86 views

Differentiability/continuity of piecewise defined functions

Let $$f(x)=\begin{cases}x^2\sin(\frac{1}{x}), &x\not= 0,\\ 0, &x = 0.\end{cases}$$ Since I can differentiate both parts of this, technically, $f$ is differentiable for all $x$. However I have ...
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0answers
29 views

Product of discontinuous and smooth function

Let $f$ be a smooth function (i.e. $C^\infty(\mathbb{R}^n)$), $g$ discontinuous in some $x_0\in \mathbb{R}^n$ and smooth everywhere else, for example $g(x)=\Theta(x)$ or $g(x)=\frac{1}{x}$. ...
0
votes
1answer
66 views

Show Open/Closed for a Set and two continuous functions

Let f, g : X → R be two continuous functions defined on a metric space X. (i) Show that the set U = {x ∈ X : f(x) > g(x)} is open in X. (ii) Show that the set F = {x ∈ X : f(x) ≥ g(x)} is closed in X. ...
0
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1answer
45 views

Two-variable limit problem: the limit of $(\cos^2(\sqrt{x^2+y^2})-1)/(x^2+y^2)$ as $(x,y)\to 0$

What is the value of $k$ such that $f$ is continuos in $(0,0)$? $$f(x,y) = \begin{cases} \dfrac {\cos^2\left(\sqrt{x^2+y^2}\right)-1}{x^2+y^2}, & \text{if $(x,y)$ $\ne$ (0,0)} \\[2ex] k, & ...
1
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3answers
46 views

continuous function

$$g(x) = \left\{\begin{array}{cl} x\sin\left(\frac{\cos(x)}{x}\right) & \text{if } x \neq 0\\ 0 & \text{if } x=0\end{array}\right.$$ Show that this function is continuous at $x=0$. so the ...