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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327 views

Show that the upper envelope of a bounded function is upper semi continuous directly

Definition 1: A real valued function $f$ is said to be upper semicontinuous at a point $p$ if: $$f(p) \geq \limsup_{x \rightarrow p} f(x) $$ Definition 2: Let $f$ be a bounded real valued ...
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1answer
118 views

$O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb R).$

Let $M(n,\mathbb R)$ be endowed with the norm $(a_{ij})_{n\times n}\mapsto\sqrt{\sum_{i,j}|a_{ij}|^2}.$ Then the set $O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb ...
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69 views

Continuity of this function(homework)

Here is a homework problem I am having trouble with: If $$f(x) = \frac{\sin{3x} +a\sin{2x} + b\cos{x}}{x^3}$$ is continuous at $x=0$, find the values of $a$ and $b$. I noticed I cannot apply ...
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1answer
65 views

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous.

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous. ($M(n,\mathbb R)$ is identified with $\mathbb R^{n^2}$ as a normed liner space.) ...
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122 views

Showing $\mathcal{H}$ is a hilbert space.

So this is an early exercise in Conway's A Course In Functional Analysis. I'm trying to get to grips with this upto open mapping and closed graph to see if I want to do any more functional analysis. ...
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2answers
50 views

Identical to a continuous function a.e.

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function. Are the following two statements equivalent to each other? $f$ is continuous almost everywhere $f$ is identical to a continuous ...
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1answer
79 views

Problem involving Properties of continuous functions

I am given two real functions $f$ and $g$ that are continuous on the interval $[a,b]$ such that for all $x$ in $[a,b]$, we have: $f(x)< g(x)$. The question is to prove that there exists a number ...
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1answer
177 views

Preimages of Jordan-measurable sets

When is the preimage of a Jordan-measurable set Jordan-measurable? In particular, is continuity sufficient? Piecewise continuity with finitely many pieces?
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1answer
285 views

Is there an unbounded uniformly continuous function with a bounded domain?

I tried to solve it by cases: domain is a set of numbers; domain is an interval,;domain is a union of numbers and some intervals. For the first case, I thought about arctanh is unbounded but its ...
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1answer
90 views

Proof involving uniform continuity

Let $f\colon [a,b] \rightarrow \mathbb{R}$ be a continuous function. Prove that the function $g$, defined by $g(x)=\max \{f(t); a \leq t \leq x \}$ is uniformly continuous. My attempt at a proof: ...
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3answers
127 views

Fixed point of continuous function

Let $f:[0,5] \rightarrow \mathbb R$ be continuous where $f(0) = f(5)$. Then $\exists c \in [0,4]: f(c) = f(c+1)$. My first idea was to show that $h:[0,4] \rightarrow \mathbb R: x \mapsto f(x)-f(x+1)$ ...
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2answers
262 views

Spaces with the property: Uniformly continuous equals continuous

I found a nice book about functional analysis with a nice theorem in it: Continuity at 0 is equal to Lipschitz continuous for linear maps in normed spaces. This fact inspires me to ask: Are there ...
2
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2answers
438 views

uniformly continuous versus continuous [duplicate]

It's hard to understand the difference between uniformly continuous function and continuous function. So if A is a uniformly continuous function on X and if B is a continuous function on X, the ...
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2answers
114 views

Continuity and product topology

I am not sure about the following. Let us consider the space of positive bounded sequences and a functional $(x_i)\rightarrow \sum_{i=1}^{\infty}\beta(i) x_i$, where $\beta$ is a function $\beta: N ...
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1answer
141 views

Showing limit of a derivative is finite

Given that a function $f$ is continuous on interval$\left[a, b\right]$, and that its derivative is finite everywhere on that interval except possibly at $c$. I am also given that $lim_{x \rightarrow ...
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4answers
1k views

Continuity on open interval

A function is said to be continuous on an open interval if and only if it is continuous at every point in this interval. But an open interval $(a,b)$ doesn't contain $a$ and $b$, so we never ...
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0answers
979 views

Continuity on open and closed intervals

I will be taking Calculus I soon, and I just want to make sure I understand some concepts correctly. So far, reading my book for Calculus I, I've encountered the definition of continuity as being ...
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112 views

continuity of the derivative under certain conditions

I am working on this exercise in a book which asks to prove that $f$ is differentiable if $f$ is continuous and that $\lim \limits_{x\rightarrow x_0} f'(x)$ exists. I know that this is easy to show ...
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1answer
191 views

Nets, dense subsets and continuous maps

Let $X$ and $Y$ be topological spaces, with $Y$ regular. Consider a dense subset $D\subset X$, a continuous map $f:D\rightarrow Y$, and a map $g:X\rightarrow Y$ (i.e. $g$ is not assumed continuous). ...
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0answers
178 views

How to fulfill those boundary conditions?

The problem is the following: Think of a set of functions depending on spherical coordinates given by: $${\psi}_{l m}(r,\theta,\phi) ={k_l}(ar) P_{l}^{m}(\cos \theta) e^{\pm i m\phi} ,$$ so ...
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1answer
641 views

Solution of $y'+2y=g(t)$, $y(0)=0$, for a given $g(t)$ having a jump discontinuity

The differential equations book that I'm reading states the following: Linear differential equations sometimes occur in which one or both of the functions $p$ and $g$ have jump discontinuities. If ...
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4answers
978 views

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
147 views

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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137 views

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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43 views

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
397 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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197 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
216 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
287 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 $ ...
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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 ...
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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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85 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 ...
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313 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
279 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 ...
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898 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 ...
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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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829 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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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
383 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 ...
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1answer
101 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 ...
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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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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 :)