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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Upper and Lower Hemicontinuity [on hold]

Let $f$ be a correspondence from $A \subset R$ to a compact set $Y \subset R$, $f: A \to R$, and define $f(x) = 1$ if $x \leq \frac{1}{2}$, and 2 if $x \geq \frac{1}{2}$ (Take note of inequalities, ...
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Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...
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23 views

How can I get the mean value of the variables with different time steps?

I am trying to get the mean value of the variables with different time steps. For example, I am trying to get the mean value of x at time t+dt which is E[x(t+dt)] as: ...
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1answer
29 views

Proving IMVT using delta-epsilon

Let's assume $f(a)<0$ and $f(b)>0$. IMVT claims that there's $c\in(a,b)$ such that $f(c)=0$. The Proof: Consider $$A = \{ a\le x\le b : f(x) < 0 \}$$ That's a non-empty set and therefore, by ...
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1answer
19 views

Continuity of a map from the 2-plane.

Let $f: \mathbb{R}^{2} \rightarrow X$ be a map where $X$ is a Hausdorff topological space. Assume that the restriction of $f$ on $\mathbb{R}^{2}-\{0\}$ is continuous, and the restriction of $f$ on any ...
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1answer
22 views

continuous on $[0,\infty)$ and uniformly continuous on $[a, \infty )$ for some $a\in \mathbb R^+$ , to show uniform continuity on $[0, \infty)$

Let $f:[0, \infty) \to \mathbb R$ be a continuous function which is uniformly continuous on $[a, \infty )$ for some $a\in \mathbb R^+$ , then how to show that $f:[0, \infty) \to \mathbb R$ is ...
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1answer
46 views

Continuity of a piecewise constant function

A)I can draw the graph and see that the function is continuous at x=0.3 as when you approach it from the left and right you get the same result B) not sure how to prove properly but it is not ...
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2answers
41 views

If a continuous function is positive at a point, it is also positive in some neighborhood of the point [on hold]

Suppose that $f:\mathbb{R}^k\to\mathbb{R}^1$ is a continuous function and that $f(x^*)>0$. Show that there is a ball $B=B_\delta(x^*)$ such that $f(x)>0$ for all $x\in B$.
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1answer
26 views

A “repeated roots allowed” version of the continuity of roots

Let $R_n$ denote the set of all monic real polynomials of degree $n$ all of whose roots are real. Then $R_n$ is a closed subset of the $n+1$-dimensional space ${\mathbb R}_n[X]$. For $P\in R_n$, ...
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4answers
109 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 $x=c$ ...
3
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1answer
27 views

Solution of differential equations with discontinuity

Suppose that we have scalar differential equation \begin{equation} \dot{x}(t)=u(t) \end{equation} Here $u(t)$ is a piecewise constant function with discontinuity. If the points of discontinuity is ...
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2answers
47 views

Conditions of Continuity (Limits)

On a math test, for my online Honors Pre-Calculus course, that I recently took I got this question wrong and don't understand the explanation: Suppose $f(x) = \begin{cases} x^2-2, & x \not= 2 ...
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46 views

Show that the function $f(\textbf{x}) =|\textbf{x}| $ is continuous on $\mathbb{R}^n$

I can see this intuitively, but looking for a solid answer with reasoning. all ideas will be appreciated,
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226 views

Continuity of piecewise function

$$f(x,y) = \begin{cases} \dfrac{\sin(xy)}{xy} & \text{if $x y \ne 0$} \\ 1 & \text{if $xy=0$} \end{cases}$$ all ideas are appreciated i think this is non-continuous, i did by converting to ...
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1answer
17 views

Hölder continuity and uniform boundedness

Is uniform boundedness is related to Hölder continuity of a function? I mean is it necessary to prove first uniform boundeness to prove the Hölder continuity of a function? Also tell me the ...
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2answers
48 views

Why the continuity of a function on a metric space doesn't depend on metrics?

In the definition of the continuous function on a metric space, it seems to me that a continuous function depends on the metric of the given metric space. Could somebody explain Why the continuity of ...
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1answer
30 views

ODE with Laplace transform: the jump of $\dot y$

I solved this eq. using the Laplace Transform: $\ddot y+4\dot y+13 y=\delta(t-2\pi)-\delta(t-7\pi)$ The sol. is: $y(t)=\frac{1}{3} e^{2 t} (-e^{14 \pi} \theta(t-7\pi) sin(3 t)+e^{4 \pi} \theta(t-2 ...
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3answers
81 views

Function with continuous inverse is continuous?

If function $\textbf{F}^{-1}(x)$ is an inverse of function $\textbf{F}$ and $\textbf{F}^{-1}(x)$ is continuous. Is it true that $\textbf{F}(x)$ is continuous too?
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On characterization of Riesz homomorphisms on $C(X)$ space

How to prove the following: Let $K$ be an arbitrary topological space and $\pi: C(K)\to\mathbb R$ be a map with $\pi (1) = 1$. If $\pi$ is a algebra homomorphism then it is an Riesz homomorphism.
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4answers
187 views

Prove/disprove: if $\lim\limits_{ n\to\infty} f(n)=\infty$ then $\lim\limits_{ n\to\infty}f(f(n))=\infty$

Let $f(x)$ a continuous function on $\Bbb{R}$. Prove/disprove: If $\lim\limits_{n\to\infty} f(n)=\infty$, then $\lim\limits_{n\to\infty}f(f(n))=\infty,$ where the limits are taken over $n \in ...
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1answer
14 views

Definitions of hemicontinuity

can anyone see the equivalence or relation between the following two definitions of hemicontinuity that I encountered: Assume that $K$ is a closed, convex subset of Banach space $X$. Let $X^{*}$ be ...
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1answer
110 views

Give an example of a continuous function $f:R\rightarrow R$ which attains each of its values exactly three times. [duplicate]

Give an example of a continuous function $f:R\rightarrow R$ which attains each of its values exactly three times. Ed.: answered by the duplicate above Does there exist a continuous function ...
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1answer
61 views

Alternative Uniform-Continuity theorem proof by Luroth

Can please someone elaborately give the proof of Uniform-Continuity theorem ( every continuous function on a closed bounded real interval is uniformly continuous) by Luroth ? thanks in advance
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52 views

A question on the purpose of the condition on hausdorff to prove homeomorphism

This is a theorem proved in Munkres. Let $f:X\to Y$ be a bijective continous function. If X is compact and Y is hausdorff, then f is a homeomorphism. I knew Y being hausdorff which will be good to ...
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1answer
30 views

Extending a homeomorphism of the open disk to the boundary.

Let $D^2 = \{x \in \mathbb{R}^2 : ||x||\leq 1\}$ denote the closed disk and $int(D^2)$ denote its interior. If I have a homeomorphism $\ f: int(D^2) \rightarrow int(D^2)$ it is clear that it is not ...
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1answer
28 views

$f$ differentiable on $[a,b]$, but not Lipschitz

Question 11-37(d) of Spivak's Calculus, 4th ed., asks If $f$ is differentiable on $[a,b]$, is $f$ Lipschitz of order $1$ on $[a,b]$? The phrase "differentiable on $[a,b]$" is a little ...
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3answers
75 views

Is $h(x_1,…,x_n)=\sqrt{x_1^2+…+x_n^2}$ continuous?

How would I go about showing whether or not $h(x_1,...,x_n)=\sqrt{x_1^2+...+x_n^2}$ is continuous? I have shown that the partial derivatives exist everywhere except $(0,..,0)$.
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The sign of $f(x)f(x+1)$ for a continuous function $f$

This is a question I tried to solve from homework. So let $f(x)$ be continuous function. I need to prove 2 things: Prove that exist $x$ such that $f(x)f(x+1)\geq0$. It seems reasonable to me, and ...
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1answer
21 views

Is $||u||_{C^\alpha} \leq ||u||_{C^1}$ for all $u$?

We have $||u||_{C^\alpha,\Omega} = \text{sup}_\Omega |u(x)|+ \text{sup}_\Omega \frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and $||u||_{C^1} =\text{sup}_\Omega |u(x)| + \text{sup}_\Omega|\frac{du}{dx}|$ I have ...
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3answers
102 views

Exponetial map from real line to circle

Is the map $x\to e^{ix}$ from real line $\Bbb R$ to circle open? If I take any closed or half closed subset instead of $\Bbb R$ then this is definitely not open. But I'm little bit confused when ...
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2answers
55 views

determine a and b so that the function is continuous

I have an assignment where I should determine $a$ and $b$ so that the following function is continuous at $x=0$: $$f(x)=\begin{cases} 2+\ln(1+x), & x>0\\ x^2+ax+b, & x\le 0 ...
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1answer
41 views

What happen to composite of infinite number of continuous functions?

We all know that a composite of continuous functions is continuous. And this holds for any $\textbf{finite}$ number of functions. My question is what happen to infinite number of functions? Is it ...
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37 views

question on differentiable and continious function

How should the function $f(x)=x\operatorname{sgn} x$ be defined at $x=0$ so that it is continuous there? Is it then also differentiable? How should the function $g(x)=x^2 \operatorname{sgn} x$ be ...
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$\lim_{x\rightarrow 1}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^x}=\ln2$.

Prove $$\lim_{x\rightarrow 1}\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}^{x}}=\ln2.$$ Of course $$\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}}=\ln2,$$ but we can not use the Proposition : If a ...
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2answers
37 views

Uniform continuity of a subset of $R$

Let $A \subset R$ and $f:A\to R$ be given by $f(x) = x^2$. Then $f$ is uniformly continuous if $A$ is bounded subset of $R$. $A$ is dense subset of $R$. $A$ is unbounded and connected subset of ...
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1answer
596 views

Proof: X is connected

Just came from an exam and I am wondering how to prove the following: A topological space $X$ is connected if for each continuous function $f:X\rightarrow X$ there is a $x \in X$ such that ...
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1answer
145 views
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A topological function with only removable discontinuities

I've posted similar questions here and here, but no one has answered them to my satisfaction. Suppose that $f:\mathbb{R} \to \mathbb{R}$ is such that $\lim_{y\to x}f(y)$ exists for all $x$, that is, ...
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1answer
19 views

For a different proof of the boundedness theorem of continuous functions

The bounded-ness theorem of continuous functions i.e. every continuous function $f:[a,b] \to \mathbb R$ is bounded on the closed bounded real interval $[a,b]$ can be proved by Bolzano -Weirstrass ...
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31 views

Calculate Radon-Nikodym derivative in a point when it is continuous in that point

I can't solve the following exercise, even if I find it quite intuitive. Let $\nu, \mu$ be Radon measures on a metric space $(X,d)$. Suppose that: 1) $w\in L^1(X,\mu), w\geq 0$ $\mu$ a.e.; 2) $w$ is ...
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1answer
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Is a function with a random variable continuous?

I often like to fool around on graphing calculators when I am bored. A function that can be very amusing is $f(x) = rand \times sin x$ Now, on my TI-84 Plus, this looks obviously discontinuous ...
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1answer
56 views

Given a continuous function f on $[a,b]$ such that $f([a,b])\subset [a,b]$, why does there exist an $x: g(x) \lt 0$ for $g(x)=f(x)-x$?

I'm trying to prove that for a function that is continuous on $[a,b]$, with $f([a,b] \subset [a,b]$ there exists a $c \in [a,b]: f(c)=c$. If you consider a function $g(x)=f(x)-x$ the result would ...
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30 views

Covering up discontinuities to create analyticity

The floor function, $\lfloor x \rfloor$ , has a "jump" at the integers where its derivative ceases to exist. Everywhere else, its derivative is zero. Now, I wish to multiply the floor function by ...
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2answers
87 views

Continuous function from R to a compact set

I know that a continuous function maps compact sets into compact sets. My question now is, are there continuous functions $f:{\mathbb R}\rightarrow I$, with $I=[a,b]$ ($a\neq b$)?
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49 views

Proof of the rank theorem in Rudin's PMA book

I am studying Rudin's proof of the rank theorem (theorem 9.32 in Principles of Mathematical Analysis.) We have an invertible function $H(x)$ defined on an open set. He claims we can "shrink" the open ...
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1answer
53 views

Prove that a function is continuous at x =0

I need to prove that $f$ continuous at $(x)=0$ using a $\epsilon$- proof $$ f(x) = \begin{cases} x/(1-x),&x\geq 0 \\ x/(1+x),&x \leq 0 \end{cases} $$ So this is what I have so far: Let ...
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1answer
18 views

Continuity along different spaces

1) Say I have a function that is continuous along $\mathbb{R}.$ Would that function be then continuous along $\mathbb{Q}$ ? How about the other way around? 2) If I have two functions that are not ...
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53 views

Examples of continuous non-transitive group actions

In studying topology, I encountered this problem: Let $S$ be a topological space and let $G$ be a topological group acting continuously on $S$ (group action as $G \times S \to S$ map is continuous). ...
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94 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
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26 views

Proving $\cos$ is Lipschitz continuous with $L=\frac{\sqrt3}2$ on $[-\frac12,1]$, using $\frac{\sqrt3}2=\cos\frac\pi6=\sin\frac\pi3$

I'm working my way through some analysis exercises to gain a better understanding and I stumbled upon an exercise where I could really use a hint. The task is to show that the inequality $|\cos ...
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2answers
50 views

Show that the graph of $y=x^3\sin(\pi/x)$ extends to a smooth arc

Here's the problem: Let $y(x)$ be a real-valued function defined on the interval $x\in [0,1]$ by means of the equation $$y(x)= \left\{ \begin{array}{lr} x^3\sin(\frac{\pi}{x}) ...