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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Proving $f(x,y) = y - x$ is continuous

How do you prove $f(x,y) = y - x$ is continuous? The domain is $\mathbb{R^{2}}$ and the codomain is $\mathbb{R}$. Is there an easy way to do it using the definition that the preimage of an open set ...
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1answer
13 views

Absolute continuity of two-dimensional measures

Absolute continuity has two different meanings: one for functions and one for measures. The Wikipedia page explains the relation between the two notions in the following way: A finite measure μ ...
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0answers
9 views

Absolute continuity for non-measures?

Let $B$ be the collection of Borel subsets of $R^2$. A measure on $B$ is said to be absolutely continuous with respect to area if any subset with area 0 has measure 0. Is there a natural ...
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0answers
18 views

Lipschitz continuity power type function [duplicate]

Is the function $f(x)=x^{\gamma+1}$, where $x>0 $ and $\gamma<0$ Lipschitz continuous ? I am a bit confused !
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1answer
54 views

If the product of two continuous bounded positive functions tends to $0$, does it follow that one of them tends to $0$? [on hold]

Let $f$ and $g$ be two continuous, bounded and positive functions on $\mathbb{R}^+.$ Given that, $\lim_{x \rightarrow \infty} f(x) g(x) = 0.$ Then prove that, at least of the functions converges to ...
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206 views
+100

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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0answers
19 views

Applications of Continuity and Differentiability on a Tough Qn

Given f is cont on [0,1] and that it is twice differentiable on (0,1). Suppose that Integral from 0 to 1 of f(x) dx = f(0) = f(1). Prove that there exist a number c where c is an element of (0,1) ...
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2answers
48 views

Uniqueness of continuous extension from $A$ to $\overline{A}$ for maps into a Hausdorff space

I want to prove the following. Let $A$ be a subset of $X$. Let $f:A \to Y$ be continuous. Let $Y$ be Hausdorff. Show that if $f$ can be extended to a continuous function $g:\overline{A}\to Y$, ...
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0answers
40 views

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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1answer
31 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 ...
4
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1answer
286 views

The product of a uniformly continuous function and a bounded continuous function is uniformly continuous

Suppose we have a bounded continuous function $f(x)$ on some interval (a,b). Suppose we also have an function $g(x)$ that is uniformly continuous on the same interval (a,b). Then, is the product ...
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1answer
50 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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4answers
508 views

Spivak's Calculus (Chapter 5, Problem 41): Proof that $\lim_{x \to a} x^2 = a^2$

In Chapter 5, Problem 41, Spivak provides an alternative way to prove that $$\lim_{x \rightarrow a} x^2 = a^2\,\,,\,\,a > 0$$ Given $\,\epsilon > 0\,$ let $$\delta = \min\left\{\sqrt{a^2 + ...
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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 ...
1
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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

Are there standard parameters for the Weierstrass nowhere differentiable function?

On Wikipedia, the Weierstass non-differentiable function is defined as: $$f(x)=\sum^{\infty}_{n=0}a^n\cos(b^n\pi x)$$ where $0<a<1$, $0<b$, and $ab>1+\frac 32 \pi$ Since it seems like, ...
1
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2answers
42 views

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

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
110 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 $x=c$ ...
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2answers
154 views

If a continuous real function is additive, then it is linear

I have to prove the following problem Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x + y) = f(x) + f(y),\ \forall x,y \in \mathbb{Z}$. Then $f$ is a linear ...
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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 ...
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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1answer
66 views

Showing if $f_n \to f$ uniformly and each $f_n$ has at most $10$ discontinuities, then so does $f$

Suppose that $f_n:[a,b] \to \Bbb R$ and $f_n$ uniformly converges to $f$ as $n$ goes to infinity. How to prove that if each $f_n$ has at most ten discontinuities (the discontinuities for each $f_n$ ...
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2answers
47 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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2answers
229 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
21 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
31 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 ...
4
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1answer
477 views

If $f$ is twice differentiable and $f(2^{-n}) = 0 $, for all $n \in \mathbb N$, then $f^\prime(0) = f^{\prime\prime}(0) = 0$.

Let $f : \mathbb R \to \mathbb R$ be a twice differentiable function, such that $f(2^{-n}) = 0$, for all $n \in \mathbb N$ . Show that $$f^\prime(0) = f^{\prime\prime}(0) = 0.$$ My attempt. First, ...
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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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0answers
23 views

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 ...
2
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1answer
111 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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3answers
425 views

How does one verify the Intermediate Value Theorem?

The Intermediate Value Theorem has been proved already: a continuous function on an interval $[a,b]$ attains all values between $f(a)$ and $f(b)$. Now I have this problem: Verify the Intermediate ...
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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 ...
2
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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 ...
1
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1answer
63 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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2answers
53 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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2answers
241 views

How to find the points at which a piecewise defined function is continuous?

Define $$ f(x) = \begin{cases} 11 & 0 \leq x \leq 1\\ x & 1< x \leq 2 \end{cases}$$ At what points is the function $f:[0,2]\to \mathbb{R}$ continuous? I am pretty sure that the ...
2
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1answer
32 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 ...
2
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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 $(0,..,0)$.
3
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3answers
48 views

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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146 views

Topology: continuous topological spaces

Let $X,Y,Z$ be topological spaces. Let $f:X\to Y$ be continuous, and let $g:Y\to Z$ be continuous. Prove $g\circ f:X\to Z$ is continuous. Use this fact to provide a detailed proof that homeomorphism ...
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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 ...
3
votes
1answer
126 views

Prove that the function is uniformly continuous

Let $f(x)$ be a continuous function in $[0,\infty)$ there are $a,b \in \mathbb{R}$ such that $\lim_{x\to\infty} [f(x) - (ax +b)] =0$ prove that $f(x)$ is uniformly continuous in ...
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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 ...
2
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2answers
256 views

Prove or give a counterexample to the following converse of theorem: A continuous function on a compact set K(subset R) is uniformly continuous.

I think the converse of this theorem is: if every continuous function over $K$ is uniformaly continuous, then $K$ is compact. To find a counterexample of it, I want to show there exist a continuous ...