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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Is it always true that if $f : D\rightarrow‎ R$ is uniformly continuous then f is bounded?(edited version)

Suppose that $D$ is a bounded set (not necessarily interval). Is it always true that if $f : D\rightarrow‎ R$ is uniformly continuous then $f$ is bounded? Prove or find counterexample. This problem ...
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321 views

How to show that a limit cannot be another number?

Let: $$ G(x) = \left\{ \begin{array} {cc} x \sin \frac{1}{x} , & x\neq 0 \\ 0, & x=0 \end{array} \right. $$ I can understand that the function is continuous at $x=0$ because: For ...
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2answers
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Continuity of $L^1$ functions with respect to translation

Let $f\in L^1$, consider the map $t\mapsto f_t=f(x-t)$, then how can one show that $t\mapsto f_t$ is continuous? More explicitly one wants to show that $\lim_{h\to 0}|f_{t+h}-f_t|_{L^1}=0$. I tried to ...
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continuous mapping is determined by its values on a dense subset of its domain

Question: If f and g are continuous mappings of a metric space X into a metric space Y, let E be a dense subset of X. if g(p) = f(p) for all p $\in$ E, prove that g(p)= f(p) for all p$\in$ X. Answer: ...
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Let $\sum_{n=1}^{\infty}n^5(\frac{x}{x+2})^n=S(x)$. Prove that the sum S(x) is a function and continuous to $x\epsilon [0,10]$

Let $\sum_{n=1}^{\infty}n^5(\frac{x}{x+2})^n=S(x)$. Prove that the sum S(x) is a function and continuous to $x\epsilon [0,10]$ Since we are talking about sums and we need to prove continuous i ...
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1answer
187 views

Zeroes of a continuous function on a metric space

Let $f$ be a continuous real valued function on a metric space $X$. Let $Z(f)$ be the set of all $p\in X$ such that $f(p)=0$ $\text{(a)}$ Prove that $Z(f)$ is closed. $\text{(b)}$ Recall ...
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1answer
393 views

Continuity of greatest integer function

Define $$ f(x):x \rightarrow[[x]]. $$ Prove that $f$ is continuous if $x\notin \mathbb{Z}$. Please use the $\epsilon -\delta$ definition of a limit. Note: I understand why does it happens.Need to ...
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2answers
196 views

Axiom of Choice, Continuity and Intermediate Value Theorem

I am trying to understand a proof I read in Herrlich's book Axiom of Choice. For those who know the book, it is theorem 4.54 on page 74. The part I am interested in reads: (9) A function $f:X ...
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3answers
214 views

continuous onto map from $(0,1)\to (0,1]$

I need to know whether There exists any continuous onto map from $(0,1)\to (0,1]$ could any one give me any hint?
3
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1answer
27 views

function with minumum in geometric mean

I have two real constants (in my case 3 and 15). I need a function that has minimum in the geometric mean and rises to infinity as I come closer to the end points. It only needs be defined on (3, 15). ...
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1answer
285 views

Lipschitz continuity and continuously differentiable functions

I have to prove that a certain function $F(x): \mathbb{R}^m \rightarrow \mathbb{R}^n$ is continuously differentiable and its Jacobian $J(x)$ is Lipschitz continuous. Are both criteria fulfilled if $ ...
2
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2answers
76 views

Continuous Functions and Their Product

Consider two functions $f,g:R\rightarrow R$. Suppose that $f$ is continuous at $0$ and $f(0)=0$; $g(x)$ is bounded but may not be continuous at $0$. Prove that $fg$ is continuous at $0$. So ...
3
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1answer
88 views

Identity makes every matrix invertible?

I have found this in a proof and do not understand where this comes from: If A is singular, then there exists $\delta \in \mathbb{R}_{>0} \forall \epsilon\in (0,\delta): \epsilon ...
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2answers
84 views

Time continuous white noise

I am aware there are similar questions about the subject, but my question is probably much more simple. I want to know why is the expectation of the second moment of continuous time white-noise ...
17
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1answer
310 views

Calculus over $\mathbb{Q}$

The mismatch between the sensitivity of 'mathematical calculus' and the flexibility of 'real world calculus' has been bothering me a bit recently. What I mean is this: in the real world, I can trust ...
2
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1answer
98 views

Regarding Limit/continuity/convergence

let $$f_n(x)=\begin{cases} 1-nx&\text{when }x\in[0,1/n]\\0&\text{when }x\in [1/n,1]\end{cases}$$ Which of the following is correct? $\lim_{ n\to\infty} f_n(x)$ defines a continuous function ...
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1answer
278 views

Preimage of open pathwise connected set is pathwise connected

Is it a fact that under a continuous function, the preimage of an open, pathwise connected set is pathwise connected itself? I'm trying to prove that $GL_n(\mathbb{C})$ is pathwise connected, without ...
7
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2answers
891 views

True Or not: Compact iff every continuous function is bounded [duplicate]

Let $X$ be a topological space. My question is: If $f:X\to \mathbb{R}$ is bounded for all such continuous $f$, then is $X$ compact. Is is really? If $X$ is the subset of $\mathbb{R}^d$, then it is ...
3
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3answers
104 views

Every topological space $X$ has the initial topology with respect to the family of continuous functions from $X$ to the Sierpiński space.

I am currently reading about initial topologies w.r.t. the Sierpiński space, and on Wikipedia I read the following Every topological space $X$ has the initial topology with respect to the family ...
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3answers
189 views

Show that $f$ is discontinuous.

Let the sequence of function $f_{n}=\sqrt[2n+1]{x}$ (for $x\geq 0$). I've shown that it converges pointwise to $f$, that is $$\lim_{n\to\infty}f_{n}(x)=f(x)=\left\{\begin{matrix} 0 ...
3
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2answers
179 views

Differences between $C_c^\infty[0,T]$ and $C_c^\infty(0,T)$

I believe it is true that: If $f \in C_c^\infty(0,T)$, then $f(T)=f(0)=0$. $C_c^\infty(0,T) \subset C_c^\infty[0,T]$ $C^\infty(0,T) \subset C_c^\infty[0,T]$ If $f \in C_c^\infty[0,T]$, it doesn't ...
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2answers
818 views

How to prove that a continuous function is bounded on an infinite interval

Given a continuous function $f\colon \mathbb R \to \mathbb R$ and the fact that $ \lim_{x\rightarrow \infty} f(x)$ and $ \lim_{x\rightarrow -\infty}f(x)$ exist (finite), prove that $f$ is bounded. I ...
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2answers
108 views

Equivalent definition of uniform continuity at infinity

I am trying to make an equivalent definition of uniform cointiniity for functions that converge at infinity. Thank you in advance for your time. Given f: R->R such that f(x)->l when x->(+infinity) ...
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1answer
239 views

Derivative inequality for a twice continuously differentiable function.

This is a question from a past exam. I thought that this was easy, but found no way of solving it. Let $f: \mathbb R\rightarrow\mathbb R$ be continuously twice differentiable with ...
3
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1answer
375 views

Compactly supported continuous function is uniformly continuous

Let $f:\mathbb R \rightarrow \mathbb R$ be continuous and compactly supported. How can I prove that $f$ is uniformly continuous ? I was trying to prove it by contradiction but get stuck. My attempt ...
0
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1answer
99 views

Concavity in discrete domain

I have question with respect to concave functions. This came up in my research. Suppose we have a real valued function $f(x)$ which is concave in $x\in \mathbb{R}$ Let $\mathbb{N}$ be the set of ...
2
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3answers
191 views

$F(x,y)$ is continuous.

Prove that $$ f(x,y)=\begin{cases}\frac{x^3-xy^2}{x^2+y^2}&\text{if }(x,y)\ne(0,0)\\0&\text{if }(x,y)=(0,0)\end{cases}$$ is continuous on $\mathbb R^2$ and has first partial derivatives ...
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3answers
63 views

Showing the existence of a limit

Please show me the existence of the limit clearly $$\lim_{\large(h,k)\to (0,0)}\dfrac{\vert hk\vert ^{\alpha} log(h^2+k^2)}{\sqrt {h^2+k^2}} =0$$ for $\alpha > \frac12$
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2answers
88 views

How to show that $f(x,y)$ is continuous.

How to show that $f(x,y)$ is continuous. $$f(x,y)=\frac{4y^3(x^2+y^2)-(x^4+y^4)2x\alpha}{(x^2+y^2)^{\alpha +1}}$$ for $\alpha <3/2$. Please show me Thanks :)
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1answer
456 views

Prove the boundedness of a bilinear continuous mapping.

Let $X,Y,Z$ are Banach spaces and $$B:X\times Y\to Z$$ is bilinear and continuous. Prove that there exists $M<\infty$ such that $$\lVert B(x,y)\rVert \leq M\lVert x\rVert\lVert y\rVert.$$ Is ...
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3answers
334 views

How to prove that càdlàg (RCLL) functions on $[0,1]$ are bounded?

While studying the space $\mathbb{D}[0,1]$ of right continuous functions with left hand limits (i.e. càdlàg functions) on $[0,1]$, I came across the following theorem: Theorem. If $f$ is càdlàg on ...
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1answer
125 views

In which of the following cases is there no continuous function $f$ from the set $S$ onto the set $T$?

In which of the following cases is there no continuous function $f$ from the set $S$ onto the set $T$? $S=[0,1],T=\Bbb R$ $S=(0,1),T=\Bbb R$ $S=(0,1),T=(0,1]$ $S=\Bbb R,T=(0,1)$ how we solve ...
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3answers
199 views

If $f:\mathbb{R}^n \to \mathbb{R}^n$ is continuous with convex image, and locally 1-1, must it be globally 1-1?

For $f:\mathbb{R}\to \mathbb{R}$ which is continuous, being locally 1-1 implies being globally 1-1, see here. This is not true for a general mapping $f:\mathbb{R}^n\to \mathbb{R}^n$. My intuition as ...
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0answers
180 views

proving that differentiable vector function $f$ is continuous at $a$

Theorem: Let $U\subseteq \Bbb R^n$ be open and $a\in U$ Let a vector function $f: U\to \Bbb R$ be differentiable at $a$ Then prove that $f$ is continuous at $a$ Proof: We need to show that ...
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2answers
60 views

Continuity and Finding Values

Find value of a,b,c such that F is continuous on the real number system: $$ f(x) = \left\{ \begin{array}{lr} -1 & : x\le-1\\ ax^2+bx+c & : |x|<1,x\ not\ equal\ to\ 0\\ ...
11
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1answer
268 views

Showing that $\Omega$ is of class $C^1$

I have done a lot in this problem, but unfortunately it is not enough to solve it, answers or hints are very welcome. Let $B$ be a rectangle in $\mathbb R^2$ and consider $\varphi\colon ...
0
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1answer
70 views

Check the continuity of the next function $f(x)=\sum_{n=1}^{\infty}(x+\frac{1}{n^2})^n$

Check the continuity of the next function $f(x)=\sum_{n=1}^{\infty}(x+\frac{1}{n^2})^n$ I've started by doing Cauchy test to check when the sum converges: ...
0
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1answer
37 views

Checking Continuity and Differentiability

Let $f(x)=\sin|x| ,x \in(-\pi ,\pi)$ Is $f$ continuous in the interval $(-\pi ,\pi)$ ? If it is, then is it differentiable on $(-\pi ,\pi)$ ?? Is it a problem of uniform continuity ? How to solve ...
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4answers
70 views

Clarification of topological space definition in terms of neighborhoods

Let $X$ be a set. We allow $X$ to be empty. Let $N$ be a function assigning to each $x$ in $X$ a non-empty collection $N(x)$ of subsets of $X$. The elements of $N(x)$ will be called neighbourhoods ...
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2answers
381 views

Discontinuity and Differentiable Functions

Given the function $f(x) = x^2$ if $x$ is a rational number and $f(x) = 0$ otherwise, prove that $f$ is differentiable at $0$, but discontinuous at all points except $ \\0$. I have the first part ...
6
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4answers
281 views

Prove that f(x)=g(x)

Show that if $f,g:\mathbb{R}\to \mathbb{R}$ are continuous and periodic and $\lim_{x\to \infty}[f(x)-g(x)]=0$, then $f=g$
3
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1answer
4k views

If $f_n\to f$ uniformly on [a,b] and f is continious on [a,b] then $f_n$ is continious in [a,b]

Yesterday I wrote a test in calculus and had to answer the following question: Prove or contradict: if $f_n\to f$ uniformly on $[a,b]$ and f is continious on [a,b] then $\exists n_0\in\mathbb N$ ...
2
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0answers
227 views

How to prove $x^y$ is jointly continuous?

It's known that real exponentiation $x^y$ is continuous in each variable, but is real exponentiation jointly continuous in both the exponent and the base? I considering the function ...
4
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1answer
59 views

Is there a simple way to state continuity for $I$-adic topology?

Let $R$ be a commutative ring with the $I$-adic topology defined by an ideal $I$, and let $S$ be a commutative ring with the $J$-adic topology for an ideal $J$. How would you translate saying that a ...
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1answer
96 views

proof of equivalence of continuity and continuity in terms of limits of sequences

I am currently working on the axiom of choice and was looking for easy applications. A common example is the proof of the equivalence of continuity with continuity in terms of limits of sequences. ...
2
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2answers
66 views

Lipschitz Continuity Contradiction

Show that $f\colon (0, \infty) \to\mathbb R$ defined by $f (x) := \frac1x$ is not Lipschitz continuous. I have the following: $$\left|\frac1x - \frac1y\right| = \left|\frac{y-x}{xy}\right| = ...
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1answer
79 views

$f(x) = x^2$ for $\Bbb{Q}$(rational) and $3(x^2)-2$ for $\Bbb{R-Q}$(irrational).Check differentiability at $x=\pi$. [closed]

Let $f$ be the following function, $$ f(x) =\begin{cases} x^2 & x\in\Bbb{Q}\\ 3(x^2)-2 & x\in\Bbb{R\setminus Q}\end{cases}$$ Check differentiability at $x=\pi$. Please help soon.
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3answers
134 views

Lipschitz Continuous Polynomials

Let f : R → R be a polynomial of degree d ≥ 2. Show that f is not Lipschitz continuous. Where exactly should I begin with this problem?
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0answers
120 views

Uniformly continuous on subsets

Let $f\colon S\to\Bbb R$ be uniformly continuous. Let $A\subset S$. Show that the restriction $f\mid _A$ is uniformly continuous. I have the following: There exists an $\alpha > 0$ such that ...
2
votes
4answers
810 views

Is function $y=\tan x$ uniformly continuous in the open interval $(0,\pi/2)\;?$

Determine whether the function $y=\tan x$ is uniformly continuous in the open interval $(0,\pi/2)$. I tried approaching it this way Let $x,y \in (0, \pi/2)$. Then $$|f(x)-f(y)|=|\tan x-\tan ...