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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Let $f$ be injective and discontinuous at some point $c$. Can its inverse be continuous?

$f$ is injective at an interval $[a,b]$, but discontinuous at some point $c$ in the same interval. I need to prove that its inverse is continuous at that interval. Should I consider what is the ...
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2answers
24 views

Let f be continuous. By EVT there exists a c such that f(c)=supx f(x). Show that f is not injective.

I am given a continuous function f in an interval [a,b]. To show that f is not injective, should I consider the definition of the extreme value theorem? I am not sure how to show that it is not one ...
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4 views

Continuity of utility function in normal form games

I want to characterize the utility functions of normal form games. Let $G$ be a game with a finite number of players $k$ given by the action sets $S_1,\ldots,S_k$ and utility function $u:S_1\times ...
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$A,B$ closed subsets of $\mathbb R^n$ , when can we say (other than compact-ness of $A$ or $B$ ) $\exists b \in B$ such that $dist(A,B)=dist(b,A)$ ?

Let $A,B$ be disjoint closed subsets of $\mathbb R^n$ , when can we say ( weaker than compact-ness of $A$ or $B$ ) that there exist $b \in B$ such that $dist(A,B)=dist(b,A)$ ? I know that if $A,B$ are ...
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39 views

CALCULUS: Sketching a function by given conditions [on hold]

Pls help. I'm currently on a struggle with this calculus problem. Thanks in advance.
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24 views

Discontinuous parametric integral function

Is there an example of a function $f:[0,1] \times [0,1] \to \mathbb{R}$ such that for all $x \in [0,1]$ the function $\phi(y) = f(x,y)$ is continuous in $y$ and for all $y \in [0,1]$ the function ...
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2answers
28 views

Show that $\langle\cdot,\cdot\rangle : E \times E \to \mathbb{R}$ is a continuous function

Let $E$ a normed vector space, where the norm is induced by a dot product. The norm of $E \times E$ is defined as $||(x,y)|| = \max\{||x||,||y||\}$. Show that $\langle\cdot,\cdot\rangle : E ...
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1answer
21 views

Proving a norm is lipschitz

Let $M\in\mathbb{R}^{n\times n}$. Define the function $f\colon\mathbb{R}^n\to\mathbb{R}$ by $f(x)=\Vert Mx\Vert$. Show that $f$ is Lipschitz. Let $x,y\in\mathbb{R}^n$, then we want to find a ...
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21 views

quotient of two differentiable functions is differentiable

I have two functions $k(t)$ and $l(t)$ in a certain closed interval $[a,b]$ both functions are continuous and differentiable in the interval. In addition we have: Both functions are increasing with ...
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1answer
31 views

conditions for continuous function

A function $f\colon [0,1]\to [0,\infty)$ is continuous and satisfies $f(0) = \lim_{x\to 0^+}\frac{f(x)}{x}$ und $ f(x)\le\int_0^x \frac{f(s)}{s}ds$ for all $x\in[0,1]$. I'm curious if it implies ...
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1answer
19 views

Continuity of the function $f(x)=\lim\limits_{n \to \infty}\frac{x}{1+(2\sin(x))^{2n}}$

I was studying the continuity of the function: $f(x)=\lim\limits_{n \to \infty}\frac{x}{1+(2\sin(x))^{2n}}$ I understood that the function behave as $ f(x)=x \quad2 \sin(x) \leq 1 \\ f(x)=0 ...
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1answer
26 views

Is this operator a distribution?

Is this operator: $$T: \mathcal{C}^{\infty}_0 \ni \varphi \to \lim_{x \to \infty} x^2 e^{-x} \varphi'(x) \in \mathbb{R}$$ a distribution (generalized function)? I need to check two things: whether ...
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1answer
16 views

How to show map is non-singular

Let $f:\;\mathbb{R}^n\to\mathbb{R}^n$ be differentiable. Suppose that for all $x\in\mathbb{R}^n:$ $$\lVert \mathrm{D}f(x)-\mathrm{I}\rVert\leq \frac{1}{2}$$ where $\lVert\cdot\rVert$ is the ...
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1answer
36 views

Proving a Function Continuous with Non-Standard Analysis

I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. ...
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11 views

Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
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1answer
52 views

Finding the domain of $\frac{1}{x}|x^2 - 1|$ [on hold]

What is the domain of this function $F(x)=\frac{1}{x}|x^2 - 1|$ Can someone please tell me how to find it ?
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3answers
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How to show that $f$ is a straight line?

Let $f:\mathbb R\to\mathbb R$ be continuous such that $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}~\forall~x,y\in\mathbb R.$ How to show that $f$ is a straight line?
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Why is the function continuous at a point which gives the case 0/0?

I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = ...
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Is it possible to extend $f(z)=\frac{\Re(z)}{|z|}$ by continuity at $z=0$?

Is it possible to extend $f(z)=\frac{\Re(z)}{|z|}$ by continuity at $z=0$? Let $z=r(\cos(\theta)+i \sin(\theta))$. Then $\frac{\Re(z)}{|z|} = \frac{r \cos(\theta)}{r} = \cos(\theta) $; as the ...
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Calculus: Proving Continuous Function by Intermediate Value Theorem [duplicate]

Prove step by step: Let $f(x)$ be a continuous function from the closed interval $[a, b]$. Use the Intermediate Value Theorem to show that $f(x)$ has a fixed point, that is, there is a point $x \in ...
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Existence of a limit - Composition of continuous functions - Questioning [duplicate]

The question of Jim Darson to this link, Don Antonio replied using a similar property in the composite of continuous functions ($\frac{\text{Re}\,z}z$ and the line $\;y=mx\;$) is continuous, but with ...
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1answer
25 views

Lipschitizianity of the square root of a positive $C^2$ function

I was trying to solve this exercise. Let $f\in C^2(\mathbb{R})$ a strictly positive function such that $f''$ is bounded. Then prove that $\sqrt{f}$ is Lipschitz. A first idea was to prove that it's ...
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2answers
51 views

Continuity and differentiability for $\sin(\sqrt x)$ & $\sinh(\sqrt {-x})$?

Let $f: \Bbb R \to \Bbb R$ with $$f(x)= \begin{cases} {\sin(\sqrt{x})\over\sqrt{x}},& \text{for } x>0\\ 1,& \text{for } x=0\\ ...
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2answers
29 views

Is it possible to extend $f$ by continuity at $z = 0$? Why or why not?

Let $f(z) = \frac{z}{|x|}$, with $z \not=0$ (a) Construct two sequences ${u_n}$ and ${v_n}$ such that $\lim_{n \to \infty} u_n = 0$ and $\lim_{n \to \infty} v_n = 0$ $\lim_{n \to \infty} f(u_n)$ ...
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2answers
514 views

If a function is discontinuous at one point, then filled in, is it now continuous?

I am looking at the continuity of the following function $f(x) = \sin(1/|x|), f(0) = 0$ So this is $f(x) = \sin(1/|x|)$ filled in at $x = 0$ Clearly, $\lim\limits_{x \to 0} f(x) = 0 $ by squeeze ...
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Can I prove a function is continuous by looking at the domain?

I came across the following question in a calculus book: For the function $$f(x)=1-\sqrt{1-x^2}$$ show that it is continuous on the interval $$-1≤x≤1$$ The solution in the book showed that the one ...
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1answer
21 views

How can I show the points of continuity of the following function

How can I show the points of continuity of the following function $$f(x) = \begin{cases} 2x, & \text{if $x \in \Bbb Q$} \\[2ex] x+3, & \text{if $x \in \Bbb I$ } \end{cases}$$ I am having ...
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60 views

Help with continuity [on hold]

Could you please clarify these questions to me. Find all the numbers for which the given function is discontinuous. $F(x)=[x-1]$ I think the solution is $\Bbb Z$ numbers right ? $F(x)= ...
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1answer
15 views

Nonlinear operator sends bounded set to relatively compact set

Consider $g$ a continuous function on $[a,b]\times\mathbb{R}$, and let $z_0\in\mathbb{R}$. Define the (nonlinear) operator on $C[a,b]$: $$Mv(x)=z_0+\int_a^x g(t,v(t))\,dt$$ for $x\in[a,b]$. Prove ...
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1answer
31 views

Let $S=[0,1) \cup [2,3]$ and $f:S \to \Bbb R$ be a strictly increasing map such that $f(S)$ is connected. Which of the following statements is true?

$f$ has exactly one discontinuity. $f$ has exactly two discontinuities. $f$ has infinitely many discontinuities. $f$ is continuous. I know theorems related to connectedness and ...
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1answer
49 views

Why is continuity needed to substitute value of derivative inside Riemann-Stieltjes Integral?

Given $f$ increasing on $[a,b]$, $g(x)\in R(\alpha)$ on $[a,b]$, $\alpha \in C([a,b])$ and $\alpha \in BV([a,b])$ $$ \beta(x)=\int_a^xg(z)d\alpha(z) \text{ on [a,b]} $$ Why is the additional ...
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42 views

Show that $f^{-1}$ is continuous

Let $E$ and $F$ two normed vector spaces, $A \subset E$ compact, $B \subset F$ and $f: A \to B$ is a bijective continuous function. As $f$ is bijective, we can defining the inverse function ...
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2answers
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Show that $Gr(f)$ is compact

Let $A \subset \mathbb{R}^n$ a compact and $f : A \to \mathbb{R}^m$ a continuous function. Let the graph of $f$ $$Gr(f) = \{(x,f(x) : x \in A)\}.$$ Show that $Gr(f)$ is compact. My proof : ...
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1answer
17 views

Definition of continuity up to the boundary

Let $\Omega \subset \mathbb{R}^n$ be open and bounded. What does it mean $f\in C(\bar{\Omega})$, i.e. what does it mean $f$ to be continuous at $x \in \partial \Omega$, maybe $$\forall \epsilon >0 ...
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Is there a nice open set proof that multiplication is continuous?

For students in a first course in analysis or topology, proving that certain function are continuous can be very tricky. However, some proofs which are difficult for students to prove using the ...
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1answer
40 views

Piece wise function continuity [closed]

Find all values of $a$ and $b$ so that the following function is continuous for all value of $x$. ($x\in\Bbb R$). $$ f(x)=\begin{cases}-3a+4x^5b&\text{when }x\le -1\\ ax-2b&\text{when ...
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2answers
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Map from circle to real line

I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that: ...
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1answer
18 views

Continuity proving of function with delta-epsilon

Prove continuity of function with the delta-epsilon definition in point $x_o=0$ $$f:\mathbb{R}\rightarrow \mathbb{R}$$ $$f(x) = \begin{cases} x^2+1, & x \in \mathbb{Q} \\[2ex] 2^x, & x \in ...
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1answer
47 views

Complex Continuity [closed]

Is the function $f$, defined by $$ f(z) = \begin{cases} \frac{z^2+2iz-1}{2z^2+iz+1} & \text{ if } z \not \in \{-i\}\\ 0 & \text{ if } z = -i \end{cases}$$ continuous at $−i$? Explain your ...
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How can I show that this function is discontinuous at the point $x=1$?

Suppose you had the function $$ f(x) = \; \text{ the integer part of } x $$ I wish to show that this is not continuous at the point $x=1$, which I will try to do by showing that $\lim_{x \rightarrow ...
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46 views

is $x\sin(\frac{1}{x^2})$ uniformly continuous on $(0,1]$?

Is $x\sin(\frac{1}{x^2})$ uniformly continuous on $(0,1]$? I am really unsure how to start this one off. I have done checks for "similar" functions like: $f=\sin(\frac{1}{x}), x\in(0,1]$, by using ...
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1answer
33 views

Show $\cos(x^2)/(1+ x^2)$ is uniformly continuous on $\Bbb R$.

now here's how I did proceed. By definition a function $f: E →\Bbb R$ is uniformly continuous iff for every $ε > 0$, there is a $δ > 0$ such that $|x-a| < δ$ and $x,a$ are elements of $E$ ...
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2answers
54 views

Is $\frac{1}{\sin x}-\frac{1}{x}$ uniformly continuous on $(0,1)$?

So I am tasked with finding whether $\frac{1}{\sin(x)}-\frac{1}{x}$ is uniformly continuous on the open interval $I=(0,1)$. To look at the "simple" ways to prove it first: I obviously can't extend ...
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23 views

Continuous function of bounded variation that is non-monotone? [closed]

Construct a continuous function of bounded variation on the interval [0,1] which is not monotone in any subinterval. We can follow the pattern of the Cantor-Lebesgue function loosely. For example, at ...
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1answer
73 views
+250

Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range

I was studying the Inverse function theorem when I came across the following problems : (Let the closed set $V$ i.e the range have non-empty interior) Does there exist a continuous onto ...
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10 views

Recovery sequence for semicontinuous functions

I have seen that the next statement holds (if my memory is not wrong) in a certain book. (I forgot which book this is.) For a lower semicontinuous function $f:(0,1)\to\mathbb{R}$ and a given ...
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2answers
29 views

differentiability of a function

Let $f$ be a continuous function on an open interval in $\mathbb R$ such that $|f|$ is differentiable. Can we show that $f$ is differentiable? I can get several examples of non-differentiable $f$ if ...
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1answer
12 views

Why is the inf-convolution of lower semicontinuous functions continuous?

I'm confusing now about the continuity of inf-convolution. I understand that the inf-convolution of lower semicontinuous functions is semiconcave and so it's locally Lipschitz continuous (in ...
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0answers
10 views

Why does $y(s)$ continuous imply that $f(s)$ with $f_l (s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum \max\{0,z_l(s)\}}$ is continuous?

Let $z:\triangle^{L-1}\to \mathbb{R}^L$ be continuous. Define $f:\triangle^{L-1} \to \triangle^{L-1}$ be defined component wise as $$ f_l(s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum_{l=1}^L ...
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0answers
17 views

Linearizing Euler's continuity equations

I have three Euler equations (for a polytropic gas) and I'd like to linearize them to get 2 other equations. Any help would be appreciated! The 3 initial equations are: 1) ...