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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Existence of solutions of the equation with a limit.

Let f be continuous function on [0,1] and $$\lim_{x→0} \frac{f(x + \frac13) + f(x + \frac23)}{x}=1$$ Prove that exist $x_{0}\in[0,1]$ which satisfies equation $f(x_{0})=0$ I suppouse that the ...
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3answers
35 views

The derivative function is not continuous

(Sorry about the bad title, couldn't think of a way to word it concisely.) Let $C[0, 1]$ be the metric space whose points are all continuous functions from $[0, 1] \rightarrow \mathbb{R}$ with the ...
3
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0answers
51 views

Why is $F$ continuous?

Why is the function: $F: P(\mathbb R) \to \mathbb R$, $F(X) = \int_X e^{-x} dx$ a continuous function? How to prove such a thing? Does it even make sense to talk about the continuity of such a ...
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3answers
20 views

Fixpoints and continuity

I don't understand why this is true: If $f:[0,1]\rightarrow[0,2]$ is a continuous function then exists $x \in [0,1]$ such that $f(x)=2x$ I don't understand why such a point exist. Why is there not ...
5
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2answers
37 views

On any continuous map $f:S^1 \to \mathbb R$

Let $f:S^1 \to \mathbb R$ be any continuous map , where $S^1$ is the unit circle in the plane . Let $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ ; then how to prove $A$ is uncountable , or ...
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1answer
14 views

Question on continuity and differentiability of min() and max() functions.

Question: $f(x)=x^2-2|x|$. Test the continuity of $g(x)$ in the interval $[-2,3]$ if $g(x)$ is defined as: attempt: $f(x)$ is defined as: But i am finding it difficult to understand $g(x)$. ...
3
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1answer
65 views

$|f(x)-f(y)| \geq \frac{|x-y|}{2}$

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $|f(x)-f(y)| \geq \frac{|x-y|}{2}$ then prove $f$ is onto. I can prove it just using IVT, but looking for some short solution which ...
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0answers
20 views

Continuous functions involving parameter [on hold]

The problem goes :Determine the parameter $a$ so that the given function is continuous at $x=1$. Draw that function. $$f(x) = \begin{cases} -x^2 + 4x - 1, & x<1\\[1ex] ax, & x\ge 1 ...
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0answers
15 views

Continuity of a parametrized surface integral of a sobolev function

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain and let $v\in H^1(\Omega)$. Furthermore, let $S=(0,T)$ denote a time interval and let $s\in ...
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2answers
84 views

Complex Analysis ( Limits at a point ).

We need to prove that $ \lim_{z \to z_{0}}(z^{2}+c)$ = $z_{0}^{2}+c$ , where c is a complex constant , using $\epsilon - \delta$ definition , where $z , z_{0}$ are complex variables. What I tried : ...
4
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1answer
95 views

Image of a continuous function

Let $f :\mathbb R \rightarrow\mathbb R$ be continuous function . Then which cannot be the image of $(0,1]$ ? A. $\{0\}$ B. $(0,1)$ C. $[0,1)$ D. $[0,1]$ Now A. is ...
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2answers
17 views

Question involving Taylor series and continuity

Question: $$f(x)=\lim_{n\rightarrow \infty}\frac{x^{2n}-1}{x^{2n}+1}$$ Where is this function continuous? Trial: I analyzed positive terms of x.For large values of n the function approaches to ...
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2answers
29 views

If $f$ is Lipschitz continuous on a closed interval $[a,b]$ such that $f([a,b])\subseteq [a,b]$ then it has a unique fixed-point

I am stucked at this problem: Prove or give a counter-example for the following sentence: If $f:[a,b]\to\Bbb{R}$ is Lipschitz continuous on a closed interval $[a,b]$ and $f([a,b])\subseteq [a,b]$ ...
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1answer
180 views

Proving existence of at least one root

The function $f:\mathbb{R}\to\mathbb{R}$, is continuous and $a>0$. How can I prove that there is at least one root of this equation: $f(x)=f(\sqrt{|x^2-a|})$
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How to prove $f:X\to Y$ is continuous

$X,Y$ are metric spaces. Then $f:X\to Y$ is continuous in $X$ if that $C\subset Y$ is closed implies that its inverse image is closed in $X$. I want a proof that's directly based on the ...
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2answers
50 views

Questions of an example of a measurable function fails to be continuous everywhere or even, almost everywhere

Definition of measurable set: A set $E$ measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a ...
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2answers
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+300

Proving the continuity of these maps

Backstory: I am having an exam soon, and these are the assignments that keep coming up, I cannot finish any of them to the end, but have ideas about solving them, and would like to hear your thoughts ...
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5answers
615 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 ...
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2answers
281 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 ...
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1answer
27 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 ...
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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 $f_n$ ...
2
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1answer
42 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, ...
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1answer
46 views

Discontinuities of an injective function from $\mathbb{R}$ to $\mathbb{R}$

It is well known that a monotonic function from $\mathbb{R}$ to $\mathbb{R}$ can have only countably many discontinuities. Question: Is it true that an injective function from $\mathbb{R}$ to ...
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3answers
153 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 ...
2
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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 ...
3
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3answers
47 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 ...
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2answers
116 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 ...
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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 ...
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2answers
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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 ...
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3answers
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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.
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0answers
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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 $\mathbb G-$ ...
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2answers
71 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 ...
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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) = ...
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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 ...
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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 ...
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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 ...
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2answers
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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 ...
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3answers
86 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 ...
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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\\ ...
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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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4answers
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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 ...
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2answers
80 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 ...
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1answer
29 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} ...
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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 ...
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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 = ...
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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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1answer
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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 ...
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0answers
15 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
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1answer
40 views

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

Using epsilon delta definition, show that $g$ is continuous on the whole of $\mathbb R$ $$g(x)=\cases{x^2 & \text{ if } x<1\\ \sqrt{x} & \text{ if } x≥1.}$$
2
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1answer
57 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) ...