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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267 views

Why formulate continuity in terms of pre-images instead of image?

I wanted to discuss my intuition of why we formulate the concept of continuity in terms of pre-image of open set is open instead of images for example if we consider $f(x) = c$ where $c$ is some ...
2
votes
1answer
40 views

Reverse Intermediate Value Theorem

What does it mean to say that a real valued function $ f : [a, b] \rightarrow \mathbb{R} $ is continuous at $ x_0 \in [a, b] $? Assume that $ f : [a, b] \rightarrow \mathbb{R} $ is continuous State, ...
1
vote
1answer
54 views

Continuity problem in derivation of general ito integral

This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ ...
1
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2answers
272 views

Continuity Must Hold in an Entire Open Set?

Claim: If a function $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at $\vec a \in \mathbb{R}^n$, it is continuous in some open ball around $\vec a$. Is this claim false? In other words, is it ...
2
votes
1answer
25 views

Uniform Continuity implies Continuity

Let $f$ be a function from a metric space $X$ to a metric space $Y$. Show that if $f$ is uniformly continuous on $X$ then $f$ is continuous on $X$. Show that the converse is not true. Uniform ...
0
votes
2answers
12 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
1
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1answer
503 views

Prove that if f is a continuous strictly monotone function defined on an interval, then its inverse is also a continuous function.

There is a theorem on continuous function that goes as follow: If f is a continuous strictly monotone function defined on an interval, then its inverse is also a continuous function. I have quite an ...
2
votes
2answers
367 views

well defined integral

Let $f$ be an increasing function on $[0,1]$ . Let $F(x)= \int_0^x f(t) dt$. I want to show integrals are well defined. My attempts: $f$ is bounded, $f(0)\leq f(x)\leq f(1)$. $f$ may only have a ...
5
votes
3answers
146 views

What is the domain of $f(x)=\frac{1}{x}-\frac{1}{x}$?

Question: What is the domain of $f(x)=\frac{1}{x}-\frac{1}{x}$? Does the function have a removable discontinuity at $x=0$? My attempt: My first intuition told me that it was $\mathbb R$, since we ...
0
votes
1answer
43 views

Is the function continuous and differentiable at $x=-2$?

The function $f: (-3, \infty) \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; -3 < x < -2 \\ 1 & \mathrm{for} \; x = -2 \\ ...
2
votes
2answers
44 views

Function on half plane, continuity

let $\mu$ be a finite positive borel measure on $\mathbb{R}$ and let $\mathbb{H}$ denote the upper half plane $\{(x,y) \in \mathbb{R}^2: y > 0\}$. consider the functions ...
1
vote
1answer
230 views

What is the difference between the terms smooth, analytical and continuous?

I saw the following (“roughly speaking”, like the author says) definition of a Lie group in ‘Group theory in Physics’, by Wu-Ki Tung: “Roughly speaking, a Lie group is an infinite group whose ...
0
votes
2answers
54 views

If a continuous function on $\mathbb{R}$ attains an extremum at a single point, it must be the global extremum.

Let $f$ be a continuous function on $\mathbb{R}$ which attains a local maximum at ${{x}_{0}}$. Prove that if $f$ doesn't have any other extremum points, then ${{x}_{0}}$ is the global maximum of $f$ ...
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votes
2answers
298 views

Rudin's definition of continuity in terms of pre-images (inverse images). Is this simple function continuous or not?

I am reading W. Rudins book ``Principles of Mathematical Analysis''. I find it hard to exactly understand the definition of continuity in terms of pre-images. Rudins definition of a continuous ...
7
votes
2answers
115 views

Proving that the product of two numbers (in $\mathbb{R}$ or $\mathbb{C}$) is a continuous function.

This is what is given in the textbook, I will highlight what is confusing me: Product in field $\mathbb R$ or $\mathbb C$,on $X \times X$ defined as: $$(x,y)\mapsto xy$$ (Let indicate that map with ...
3
votes
3answers
45 views

How to show $\sqrt{|x|}$ is not Lipschitz continuous?

$f(x) = \sqrt{|x|}$ is a famous example of a function which is not Lipschitz continuous but is uniformly continuous. This link shows detailed explanation of it. Here provides the figure of this ...
1
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1answer
41 views

“continuously differentiable $\subseteq$ Lipshitz continuous” with $f(x) = x^2$

In the Wiki, it says: continuously differentiable (i.e. class $C^1$) $\subseteq$ Lipshitz continuous. Consider the simplest example ($x,y\in \mathbb{R}$): $$f(x) = x^2$$ It is not Lipshitz ...
2
votes
2answers
29 views

Function of several variables which is continuous at single point

Examples of functions on $\mathbb{R}$ which are continuous at a single point are well known. But what about $f:\mathbb{R}^2\to \mathbb{R}$ which is continuous at a single point? I tried to proceed as ...
0
votes
3answers
49 views

Proof that |x-a| is continuous at x=a (epsilon delta), and nondifferentiable at x=a.

I need help justifying that $|x-a|$ is continuous and non-differentiable at $x=a$. I would also like to prove that it achieves a minimum at $x=a$, but I do not know if that is already clear enough.
0
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1answer
35 views

Prove that a function is continuous using basic open sets

Using basic open sets of $\Bbb R$, prove that $f(x,y,z)=x^2+y^2+z^2+2x+2y+6$ is a continuous function from $\Bbb R^3$ to $\Bbb R$. My attempt: Since $f(x,y,z)$ is continuous and $f(x,y,z)\in ...
2
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0answers
25 views

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where…

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where $$\mathbb{I_r}=[x_0-r,x_0+r]$$ and $$\mathbb{P}=\{(x,y): |y-y_0|\leq a, |x-x_0|\leq b\}\subset \mathbb G $$ where $\mathbb G-$ ...
3
votes
2answers
69 views

Why do we care if a function is uniformly continuous? [duplicate]

There are a lot of question regarding whether a function is or is not uniformly continuous or just continuous and there are a lot of $\epsilon_s$ and $\delta_s$ trying to show whether a function is ...
1
vote
1answer
41 views

Decide if the following functions are not continuous on $(-\infty, \infty)$

Suppose $g(x)$ is continuous on $(-\infty, \infty)$. Determine if the following functions are or are not cont. on $(-\infty, \infty)$ and explain. a) $k(x) = \frac{x^2}{4 - (g(x))^2}$ b) $j(x) = ...
2
votes
1answer
42 views

Non injective continuous maps

Motivated by comments on this question we ask the following question: Let $f:M\to M$ be a continuous map where $M$ is a compact manifold and $f$ is not injective. Are there necessarily ...
1
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1answer
44 views

A continuous function from $\mathbb R\to \mathbb R^2$

Is there a continuous function $f:\mathbb R\to \mathbb R^2$ such that $f(\cos n)=(n,\frac{1}{n})$ for all $n\in \mathbb N$? I think this is not possible as if $f$ is continuous then the function ...
0
votes
4answers
74 views

Proving that a continuous $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A) \text{ connected}$

Proving that $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A)-\text{connected}$ The answer is given like this just one step I do not understand ...
1
vote
1answer
318 views

Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous

Working on the following problem from Munkres: Let $(X, d_{X})$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f:A \to Y$ is uniformly continuous, then ...
0
votes
0answers
36 views

The space of continuous functions as a dual space

Let $X$ be some topological Hausdorff space and $C_b(X)$ the space of bounded complex continuous functions on $X$. Is there a Banach space $B$ such that $B^* \simeq C_b (X)$? I know of a very similar ...
1
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2answers
58 views

Why is $f(x) = x^2$ uniformly continuous on [0,1] but not $\mathbb{R}$

According to How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity? There is a lot of agreement that $x^2$ is not uniformly continuous. But is $x^2$ uniformly ...
6
votes
3answers
85 views

Prove that $f'(0)=L$.

Let $f$ be continuous at $0$. Suppose lim$\displaystyle _{x\rightarrow 0} \frac{f(2x)-f(x)}{x} =L$. Prove that $f'(0)=L$. My Work: $\displaystyle ...
1
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0answers
25 views

A question on continuity of a piecewise function with 4 constants

I have this function, and I need to find the values of $a, b, c$ and $d$ so that $f(x)$ will be differentiable everywhere. $$f(x)=\begin{cases} ax+b, & x<-2 \\ x^2+c, & -2\le x\le2\\ ...
1
vote
2answers
55 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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2answers
88 views

Homework problem on continuity

Let $U =\{A \in M_{n} : A \text{ is invertible}\}$ (where $M_{n}$ is the space of all $n\times n$ matrices). $U$ is an open subset of $M_{n}$. Define $\alpha: U \rightarrow M_{n}$ by ...
3
votes
2answers
78 views

A ring is a connected set

I not know how to prove this: For example $$A=\{(x,y,z)\in \mathbb{R^3}\mid 1 < x^2 + y^2 + z^2<2 \}$$ I know that $$\partial A=S(0,1)\cup S(0, \sqrt{2})$$ can that help me at all? I was also ...
1
vote
1answer
43 views

Show that the sequence does not converge

My Try: $|f'(a)|>1$. Assume that the sequence converges to a limit $b$. Then $f(b)=b$. Since $a$ is the only fixed point it implies that $b=a$. Hence, given any $\frac{1}{m}$ where $m\in ...
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votes
0answers
37 views

Smoothing a function [closed]

Can you smooth a non-smooth function by: Differentiating it until you get a non-continuous function Changing that derivative to make it continuous by replacing the portions where there are jumps by ...
0
votes
1answer
14 views

Continuous function space and Reproducing kernel Hilbert [closed]

Let $E=C[-1,1]$, space of all real-valued continuous functions on [-1,1], $E$ is a reproducing kernel hilbert space? by inner product $\int_{-1}^{1} f(x)g(x) w(x) dx$ where $w(x)>0$ is weighted ...
1
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1answer
28 views

Question about continuity of piecewise function of two variables

Let $$ f(x,y)= \left\{ \begin{array}{ll} \left(x\sin\left(\frac{y}{x}\right),\frac{\cos (y) -1}{y}\right) & x \neq 0 \wedge y \neq 0 \\ (0,0) & x = 0 \vee y = 0 \\ \end{array} ...
0
votes
2answers
31 views

Piecewise $\mathscr C^1$ and piecewise continuous

I'm a little bit confused in piecewise continuity of a function. Say, if we have an odd function like $f(x) = x$ defined on the open interval $(0, \pi)$. We then extend it to a period $2\pi$ function ...
7
votes
2answers
163 views

Values of $a$ s.t. for all continuous $f$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)$

Determine all $a\in[0,1]$ such that for ${\it every}$ continuous function $f:[0,1]\to \mathbb{R}$ with $f(0)=f(1)$ there exists at least one $x$ where $f(x) = f(x+a)$. First of all, $a=0,1/2,1$ ...
5
votes
2answers
141 views

continuity of a function

I have a task as preparation for my Calculus Exam. $f(x)= \begin{cases} 2^{\frac{1}{x-2}} ,& x\neq 2 \\ 0 ,&x=2 \end{cases}$ Now we have the following solution by one of our tutors: $l_1 = ...
2
votes
1answer
56 views

Continuity of norm. Need to understand how and why

$f:X \to \mathbb R \ \ \ , \ f(x)=\| x\|.$ Prove that $f$ is continuous. I have this definition of continuity in metric spaces: Let $(X, d_x)$ and $(Y,d_y)$ be metric spaces. $$f\in C(a) ...
1
vote
1answer
23 views

Showing that a continuous function is greater than zero

I've been working on a problem that wants me to show that given a function $f$ that is continuous at the point $c$ that, $$f(c)>0 \to \exists \delta\;\ \text{such that}\;\ f(x)>0\; \forall x ...
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votes
1answer
34 views

Proving that is $A:X \implies Y$ is a linear operator from metric space X to Y is continuous iff it is bounded bounded

The $\implies$ part interests me. The proof given goes like this: Let $A$ be continuous in 0 (because the 0 vector is in every vector space) $B_y(0,r)=\{y \in Y | \| y\|<r \} \implies \exists ...
0
votes
0answers
14 views

is this multivariable function twice continuously differentiable with respect to the parameter?

I have the following function $V: {R}^{*}_{+} \times {R}^{*}_{+} \rightarrow R$ , a and b are strictly positive real coefficients: $$V ( x_i(l) , x_j (l) ; l ) = a x_i (l) - b x_i (l)^2 + l^2 x_i ...
2
votes
1answer
39 views

Show continuity using epsilon delta definition for piecewise function [closed]

Using epsilon delta definition, show that $g$ is continuous on the whole of $\mathbb R$ $$g(x)=\cases{x^2 & \text{ if } x<1\\ \sqrt{x} & \text{ if } x≥1.}$$
3
votes
4answers
388 views

Derivability of a piecewise function

Let's say I have a continuous piecewise function of a single variable, so that $y = f(x)$ if $x < c$ and $y = g(x)$ if $x>=c$. Is it right to say that the derivative of the function at $x=c$ ...
1
vote
0answers
35 views

Continuity by composition with a homeomorphism

I only want to know what do you guys think about the following proof. That's an exercise I've tried to do and I don't have an available answer, so... If you find some error or imprecision, I'd be ...
2
votes
1answer
54 views

What is this subclass of the class of monotonic transformations?

Let $u$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for ...
13
votes
3answers
5k views

Prove that the function$\ f(x)=\sin(x^2)$ is not uniformly continuous on the domain $\mathbb{R}$.

If I want to prove that the function$\ f(x)=\sin(x^2)$ is not uniformly continuous on the domain $\mathbb{R}$, I need to show that: $\exists\epsilon>0$ $\forall\delta>0$ ...