Questions tagged [continuity]

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) or (general-topology)

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23 views

Is $f(x)= x$ for $x\in\mathbb{Q}$ and $f(x)=-x$ for $x\in\mathbb{R}\backslash \mathbb{Q}$ differentiable in $x_0=0$

I figured that it is continuous, but I fail to prove/disprove that it also is differentiable in $x_0=0$
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1answer
26 views

Topology and continuity from right

How do we define continuity from right on an ordered set with the order topology, using the specific definition of continuity which is inverse image of an open set is open? Thanks in advance.
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41 views

Characterization of real-valued $C^1$ functions on $[0,1]$

I have encountered the following example, where I found things I don't understand: Let $\Bbb{Q}^+$ be the set of positive rational numbers, $C([0,1])$ be the space of all continuous real-valued ...
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4answers
89 views

Compute $\lim\limits_{x\to 0^+}(1+x^2)^{\frac{1}{x}}$?

so I started the casual way:$$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^+}(1+x^2)^{\frac{1}{x}}=\lim\limits_{x\to 0^+}e^{\ln \left((1+x^2)^{\frac{1}{x}}\right)}$$ How can I proceed?
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1answer
56 views

Finding the points of continuity of $f(x) =\sum_{n=1}^{\infty} \frac{\sin nx}{n^{1/2}}$

Let $$f(x) =\sum_{n=1}^{\infty} \frac{\sin nx}{n^{1/2}}$$ How can I find the points of continuity of this function? I know that continuity of $\frac{\sin nx}{n^{1/2}}$ doesn’t guarantee the ...
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1answer
29 views

Smoothly homeomorphic for invariance of domain and invariance of dimension

Follow-up to this: Do homeomorphic smooth manifolds, like diffeomorphic ones, have the same dimension? Based on this question Viewing invariance of domain as a converse of invariance of dimension, ...
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1answer
12 views

Proof that the harmonic function defined by the Poisson integral is continuous on the boundary of a ball.

I am reading the text Elliptic Partial Differential Equations of Second Order by D.Gilbard and N.Trudinger, I am struggling with a particular proof they have given. (p.21) In the text they are ...
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1answer
47 views

For manifolds of the same dimension, are submersions equivalent to immersions?

My book is An Introduction to Manifolds by Loring W. Tu. Immersions and submersions are defined here. Let $A$ and $B$ be manifolds with the same dimension $d$, and let $G: A \to B$ be a smooth map. I ...
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2answers
109 views

Do homeomorphic smooth manifolds, like diffeomorphic ones, have the same dimension? [duplicate]

For smooth manifolds $A$ and $B$ with respective dimensions $a$ and $b$. If $A$ and $B$ are diffeomorphic, then $a=b$. I guess the same is true for homeomorphic topological ($C^0$, I guess) manifolds (...
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1answer
61 views

Proof verification. Study continuity of $f\circ g$ and $g\circ f$ if $f(x) = 1-|x-1|$ and $g(x) = x$ for $x\in\Bbb Q$, $g(x)=2-x$ for $x\in\Bbb I$

Given two functions: $$ \begin{align} f(x) &= 1 - |x-1|\\ g(x) &= \begin{cases} x,\ \text {if $x\in \Bbb Q$}\\ 2-x,\ \text {if $x\in \Bbb R\setminus\Bbb Q$} \end{cases} \end{align} $$ ...
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30 views

Notions of continuity for stochastic processes

I would like to receive some clarification regarding the difference between continuous in probability and continuous almost surely. Using the definition of the wikipedia page (that match the one I ...
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1answer
128 views

$f:[0,1]\to[0,1]$ be a continuous function. Let $x_1\in[0,1]$ and define $x_{n+1}={\sum_{i=1}^n f(x_i)\over n}$.Prove, $\{x_n\}$ is convergent

I have tried a little bit which as follows- Since $f(x_n)\in[0,1]$, $\{f(x_n)\}$ has a convergent subsequence say $y_n=f(x_{r_n})\ \forall n\in\Bbb{N}$ Let, $\lim y_n=l\implies \lim \frac{y_1+y_2+\...
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1answer
44 views

Continuity of a function at 0 [closed]

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be $$f(x)=\begin{cases}\frac{p(x)}{q(x)}& \text{ if } x\neq 0\\ 0 & \text{ if } x=0 \end{cases}$$ where $p(x)=x_1^5x_2^3$ and $q(x)=x_1^6+x_2^8$. Is $...
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1answer
28 views

Stucked at the very last step for proving additive function over the reals

I'm stucked at the very last step to prove that for any additive function over the reals, $f(x)=kx$. This is a previous question, but I don't understand how to fit continuity with the fact that $x$ is ...
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1answer
30 views

For what real values is the following function discontinuous?

If $x$ is irrational or $x=0$, we set $f(x)=0$. If $x \in\mathbb{Q}$ and $x=m/n$ such that $m,n\in\mathbb{Z}\setminus\{0\}$, $GCF(m,n)=1$, then we set $f(x)=1/n$. I find this question confusing ...
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1answer
41 views

Proof of Discontinuity: $f:\mathbb{R}\to \mathbb{R}, x\mapsto f(x)=\sin(1/x)$

Prove the Discontinuity of $f:\mathbb{R}\to \mathbb{R}: x\mapsto f(x)=\begin{cases} \sin\left( \frac{1}{x}\right) \quad \quad x\neq 0\\0 \quad \quad \quad \quad x=0\end{cases}$. My Proof: Since $\...
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3answers
55 views

Without using the IVT, prove that if $\text{Dom}f=\text{Im}f=[0,1], f$ continuous, then $f$ has a fixed point

I'd like to know if my approach is sound (also, I'd like to know if there are purely constructive proofs). Let $g: [0,1] \to [0,1], g(x)=x$. We want to prove that the equation $$g-f=0$$ has a ...
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0answers
16 views

Is $\sum_{n \in \mathbb{Z}^d \backslash\{0\}} \mathrm{e}^{\mathrm{i} \langle x, n\rangle} / \lVert n \rVert^d$ discontinuous?

I consider the following periodic function $f : \mathbb{R}^d \rightarrow \mathbb{R}$, given by $$f(x) = \sum_{n \in \mathbb{Z}^d \backslash\{0\}} \frac{\mathrm{e}^{\mathrm{i} \langle x, n\rangle} }{\...
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1answer
21 views

Preserving continuity of periodic functions under fractional-integration-type transformations

Assume that $f$ is a continuous and periodic function over $\mathbb{T}=[0,1)$, and denote by $f_n$, $n \in \mathbb{Z}$, its Fourier series. Let $(a_n)_{n\in \mathbb{Z}}$ be a sequence such that $n^a ...
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0answers
39 views

Mean value theorem for a function with a point of discontinuity [closed]

According to the definition, Let $f:[a,b]\to \mathbb {R}$ be a continuous function on the closed interval $[a,b]$, and differentiable on the open interval $(a,b)$, where $a<b$. Then there ...
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2answers
62 views

Why is $f:[a,b] \to \mathbb{R}, f(x)=x^2$ Lipschitz continuous?

Proof: Let $a,b\in \mathbb{R}$ with $a<b$. $f$ is Lipschitz continuous since $\forall x,y\in [a,b]:$ $$|f(x)-f(y)|=|x^2-y^2|=|x+y|\cdot |x-y| \overset{(*)}\leq (|x|+|y|)\cdot |x-y| \leq 2\max \{|...
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0answers
52 views

Continuity of $f(x)=\lfloor x\sin \pi x\rfloor$

If $f(x)=\lfloor x\sin \pi x\rfloor$ (where $\lfloor x\rfloor$ denotes the greatest integer that is at most $x$) is $f$ continuous at any point? Also prove whether it's continuous or not using the ...
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2answers
41 views

Topology Hausdorff space

Let $X$ be Hausdorff space and $f$ is a continuous function from $[0,1]$ to $X$. If $f$ is one-one, then image of $f$ is homeomorphic to $[0,1].$ I did something like defining mapping $g$ from image ...
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1answer
42 views

Riemann integral of non-negative continuous on $\mathbb{R}$

Everyone might get used to the following question in the context when $f$ is assumed to be Lebesgue integrable function. In this problem, we give new hypotheses Let $f:\mathbb{R}\rightarrow\mathbb{R}$...
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1answer
30 views

Find function $f'(x)$ such that its domain $D'=\mathbb{R}$, $f'(x)=f(x)$ $\forall x\in D$ and $f'(x)$ is continuous.

Let $f(x)=\frac{^3\sqrt{x^3+3x^2+7}}{x+2}$. I was asked to find $f'(x)$ such that $a)$ the domain $D'$ of $f'(x)$ is $\mathbb{R}$, $b)$ $f(x)=f'(x)$ $\forall x\in D$, with $D$ being domain of $f(...
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0answers
90 views

Is $f$ a continuous function?

I seek to understand whether a certain discontinuous function on the dyadic rationals can be recast using the Cantor set as a continuous function. Let $X$ be the dyadic rationals in the interval $(\...
11
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2answers
135 views

Study continuity of $f(x) = |x|$ for $x\in\Bbb R \setminus \Bbb Q$, $f(x) = \frac{qx}{q+1}$ for $x\in\Bbb Q$.

Study continuity of the following function: $$ f(x) = \begin{cases} |x|,\ \text{if $x$ is irrational}\\ \frac{qx}{q+1},\ \text{if}\ x = {p\over q}, q\in\Bbb N, p\in\Bbb Z, p\perp q \end{cases} $$ I'...
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2answers
81 views

Do we call $x^{\frac{3}{2}}$ discontinous and non differentiable at $ x = 0$

Do we call $x^{\frac{3}{2}}$ discontinous and non differentiable at $ x = 0$ ?? Because the function has the domain as $ x ≥ 0 $ , so we cannot have a LHL for it. So do we call the function ...
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2answers
43 views

Confused about the algebra of continuous functions

Provide an example (or explain why the request is impossible) of a pair of functions $f$ and $g$ neither of which is continuous at $0$ but such that $f(x)g(x)$ and $f(x) + g(x)$ are continuous at $0$; ...
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1answer
47 views

Very specific quibble re: proving that $g(x)=\sqrt[3]{x}$ is continuous at $c \neq 0$ using $\epsilon-\delta$ arguments

I'm mulling over an exercise in Abbott's textbook. We want to prove that $\forall \epsilon>0 \exists \delta>0$ such that, if $0<|x-c|<\delta$, then $|g(x)-g(x)|< \epsilon$. A little ...
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1answer
41 views

There exists injective homomorphism from $\Bbb R \oplus \Bbb R$ to $C(\Bbb R)$

True or False: There exists injective ring homomorphism from $\Bbb R \oplus \Bbb R$ to $C(\Bbb R)$ where $C(\Bbb R)$ is the set of continuous function from $\Bbb R$ to $\Bbb R$. I was trying to think ...
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5answers
125 views

What Can We Say About the Continuity of $y=\frac{x}{x}$ at $x=0$?

If we can't divide by $0$, should $\frac{x}{x}$ be discontinuous and undefined at $x=0$ or is it continuous with value $1$? Most online graph calculators plot a continuous curve. If it's continuous ...
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0answers
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Can someone help me with Fourier? [closed]

can someone give me a hand with the section c) and d)? I can not do it ... I think the point is that I do not understand how to integrate the function and then draw what it asks for. Problem
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1answer
176 views

If $f$ is continuous and $f'(x)\ge 0$, outside of a countable set, then $f$ is increasing

PROBLEM. Let $f:[a,b]\to\mathbb R$ be a continuous function, such that $f'(x)\ge 0$, for all $x\in [a,b]\setminus A$, where $A\subset [a,b]$ is a countable set. Show that $f$ is increasing. Attention....
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1answer
22 views

Changing the order of quantifiers in the definition of continuity of function

I wonder what if we could change the order of the quantifiers in the definition of continuity of function. I mean $a)$ For any number $\forall \delta >0,$ there exists some number $ \exists \...
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2answers
29 views

Continuity between two metric-spaces

How to prove (without sequences) that $id: (\mathbb{R},|.|)\to (\mathbb{R},d)$ is continuous? where $d(x,y)=|\exp(x)-\exp(y)|$ can we say: id is continuous iff $$\forall x_0 \in \Bbb{R}, \forall \...
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3answers
59 views

Let $X \subseteq \Bbb Q^2$. Suppose each continuous function $f:X \to \Bbb R^2$ is bounded. Then $X$ is finite.

True or false: Let $X \subseteq \Bbb Q^2$. Suppose each continuous function $f:X \to \Bbb R^2$ is bounded. Then $X$ is finite. Now it will be compact for sure just by using distance function. Now ...
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2answers
64 views

Compact subsets for a given topology

I am currently studying topology in real analysis and have a problem that I'm stuck on and really don't understand completely. Question: Consider the topology $ \tau$ = {$\emptyset$ ,$\mathbb{R}$} ...
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2answers
28 views

Showing well-definedness of an equivalence class in $L^{\infty}(\mathbb R)$

Let $\mathbb L:=\{ f\in L^{\infty}(\mathbb R): \operatorname{there exists a representative}$ $\widetilde{f}$ so that $\lim\limits_{x \to \infty}f(x):=\lim\limits_{x \to \infty}\widetilde{f}(x)\...
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2answers
41 views

Why does $(f_{n})_{n}$ equicontinuous and $f_{n}$ uniformly continuous not imply $ (f_{n})_{n}$ uniformly equicontinuous

Why does $(f_{n})_{n}$ equicontinuous and $f_{n}$ uniformly continuous not imply $ (f_{n})_{n}$ uniformly equicontinuous. I have already seen the example of the case $(f_{n})_{n}$ where $f_{n}(x)=\...
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1answer
44 views

Interest Modeling Problem Mistake?

Adriana opens a savings account with an initial deposit of $\$1000$. The annual rate is $6\%$, compounded continuously. Adriana pledges that each year, her annual deposit (deposited continuously) will ...
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0answers
12 views

Example of a noncontinuous demi-continuous operator?

Let $(E,\left \| . \right \|)$ be an ordered Banach space and $T: E\times E\rightarrow E$ an operator. $T$ is said to bemonotone demicontinuous in (x,y), if for any two monotones sequences $(x_n)...
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0answers
41 views

Differentiable functions whose derivative is continuous

I am just confused on the structure of differentiable functions whose derivative is also continuous. How will that continuous function be? What extra restrictions one need to put so as to make a ...
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5answers
99 views

Intuitively, why are the two limit definitions of $e^x$ equivalent?

Thanks for reading! Intuitively, why does... $$\lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{xn}=\lim_{n \rightarrow \infty} \left(1+\frac{x}{n}\right)^{n}=e^x$$ Note, I'm not asking why $...
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1answer
25 views

Study continuity of the function $f(x) = \lim_{t\to+\infty} (1+x)\tan(xt)$ and sketch its graph.

Given a function: $$ f(x)= \lim_{t\to+\infty} (1+x)\tan(xt) $$ Study its continuity and sketch a graph. This problem comes along with several similar ones, one of which was to study the ...
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0answers
27 views

Understanding the maximum theorem. Continuity of convex optimization problem with parameterized constraints.

Given \begin{align} s(p)= &\text{arg min}( g( \boldsymbol{x}) ) \\ \text{subject to }& \boldsymbol{A}(p) \boldsymbol{x} = \boldsymbol{b}(p) \\ &c_1 \le x_i \le c_2 , i \in \{1,2,\cdots,...
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2answers
40 views

Is there a function that has point reflection in $(0,0)$ and that is only not differentiable in $x=0$?

Is there a function that has point reflection in $(0,0)$ and that is only not differentiable in $x=0$? I don't think that this is possible, but I can't prove it - can you find an example where this ...
0
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1answer
18 views

Sequence of continous functions converges uniformly to a funtion then that function is continous

I have to following exercise and I am not sure about its solution. Let $f_h\colon A\subset \mathbb{R}^n \rightarrow \mathbb{R}$ with $h=1,2,\dots$ a sequence of continous functions and let $f\colon A\...
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1answer
28 views

Prove That a Specific Equation has a Solution Over a Given Closed Interval

how would you prove that the equation $e^x=\frac{9}{1+9\cos(x)}$ has a solution over the interval of $[0, \frac{\pi}{2}]$? Isn't it enough to just state, that $g(x)=e^x$ and $f(x) = \frac{9}{1+9\cos(x)...
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1answer
24 views

Can I say an inverse function is continuous at point by the continuity of composite function theorem?

It is very common to find in calculus textbooks the following theorem: if the functions $f$ and $g$ are continuous at $x_0$, then the composite function $f^\circ g$ function is also continuous at $...