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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A continuous mapping $f:\mathbb{R}\rightarrow\mathbb{R}$ may have a fixed point?

Let a function $f:\mathbb{R}\rightarrow\mathbb{R}, $satisfied $$\forall x,y\in\mathbb{R},|f(x)-f(y)|\leq k|x-y|.(0<k<1)$$ Prove: There exists a only one $\xi\in \mathbb{R}$ ,such that ...
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Characterization of continuity in terms of preimages of open sets

1--8 Theorem. If $A\subset \mathbb R^n$, a function $f:A\to \mathbb R^m$ is continuous if and only if for every open set $U\subset \mathbb R^m$ there is some open set $V\subset \mathbb R^n$ such ...
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52 views

Showing two functions are uniformly continuous

I have no idea how to prove this detail (uniformly continuous) about these functions because they're defined to $\infty$. I need the general mindset to prove it, or any ideas. Thanks in advance. $$ ...
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how to find the value of k

f(x)=\begin{cases} k(x^2-2x),x\le 0 \\ 4x+1,x>0 \end{cases} continuous at x=0 I am extremely weak in this topic so could any one show me how to solve this question?
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Make a multivariable function continuous

What can we do with this function, so the function will be continuous in $(0,0)$? $f:\mathbb{R}^2\rightarrow\mathbb{R}:(x,y) \mapsto \frac{x^2+y^2-x^3y^3}{x^2+y^2}$ What I think we should do, is: ...
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46 views

Continuity and differentiablity [closed]

True or False ? If $f : \mathbb R \to \mathbb R$ satisfies $$|f(x) − f(y)| ≤ |x − y|^{\sqrt{2}}$$ for all $x, y \in R$, then $f$ must be a constant function. Let $f : \mathbb R\to \mathbb R$ be ...
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1answer
31 views

Maclaurin series for $\frac{1}{|1+x|}$

I believe that there is no Maclaurin Series for $\frac{1}{|1+x|}$ as the latter is not differentiable at $x=-1$. However, would it be appropriate for me to refer $\frac{1}{|1+x|}$ as 'not a smooth' ...
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2answers
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Why must a continuous function be null if its definite integral is null? [duplicate]

Let $ f(x) = \begin{cases} f:[a,b] \rightarrow\mathbb R \\ \int_{a}^{b}f = 0 \end{cases}$. Prove: if $f$ is continuous, then $f\equiv 0$. I'm still trying to get the intuition on the situation. For ...
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83 views

2 examples to try to understand partials derivatives and deriviability

To prove that a functions has partial derivatives every partial has to exist, and every partial exist only if the limit of definition of partial exist. Is this right? Then if partials exist ,and the ...
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2answers
35 views

Proving that continuous map from subset of $\mathbb{R}^2$ is closed

Let $\delta>0$ and $f:\mathbb B((0,0),\delta)\to\mathbb R$ is a continuous map, where $\mathbb B((0,0),\delta)=\{(x,y) \in \mathbb R^2 \text{ such that } x^2+y^2\leq\delta^2\}$ Prove that there ...
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1answer
34 views

Continuity of the maximum of a function in two variables

The function $f( x, y)$ is continuous on $x\in [a,b]$, $y\in [a,b]$. Is the function $g(x) = \max_{y} f( x, y)$ continuous on $x\in [a,b]$?
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67 views

derivability don't imply partial to be continuous ? example

Is $$f(x,y) =\begin{cases} x^2+2x+2y & \text{ for } (x,y)\neq (0,0) \\ y^2 & \text{ for } (x,y)=(0,0) \end{cases}$$ derivable? But its partials are not continuous?
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1answer
20 views

Derivatives on both side of an asymptotic equivalence

Suppose I have two continuous function $x(t)$ and $w(t)$. If I have that $x(t)\rightarrow w(t)$, does that imply that $\dot{x}(t) \rightarrow \dot{w}(t)$?
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32 views

Removable discontinuity or asymptote?

The difference between a "removable discontinuity" and a "vertical asymptote" is that we have a R. discontinuity if the term that makes the denominator of a rational function equal zero for x = a ...
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27 views

continuity of bilinear

Let $B: E\times F\rightarrow G$ be a continuous bilinear map of normed spaces, where $\|(e,f)\| = \|e\| _E+\|f\|_F$. Show that $\dfrac{\|B(e,f)\|}{\|(e,f)\|} \rightarrow 0$ as $(e,f) \rightarrow 0$. ...
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34 views

continuity, discontinuity derivative and relation to being derivative but its partials are not continuous

is $$f(x) =\begin{cases} x^2\sin(\frac{1}{x}) \mbox{ for } x\neq 0 \\ 0 \mbox{ for } x= 0\end{cases}$$ a continuous function specially at point x=0? And why being derivable its derivative is not ...
2
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1answer
23 views

continuity single and multivariable function simple question

Why $$f(x,y) =\begin{cases} \frac{xy^2}{x^2 +y^2} \mbox{ for } (x,y)\neq (0,0) \\ 0 \mbox{ for } (x,y)= (0,0)\end{cases}$$ is continuous and $$f(x) =\begin{cases} 2 \mbox{ for } 0>=x>10 \\ 5 ...
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1answer
24 views

continuity single variable function and multivariable funtion and its parcial derivatives

Is f(x)=1/x discontinuous at point x=0 or not since its domain is x>0 and x<0? And what about f(x,y)=$\frac{xy^2}{x^2+y^2}$ continuity? And Df(x,y) exist or parcial derivatives are ...
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1answer
47 views

Continuity definition and theorem in a topology

This is an extremely common theorem, I have a function $f$ that maps $f:(X,\mathscr{S})\to(Y,\mathscr{T})$. I want to show that $f$ is continuous if and only if for all $V\in \mathscr{T}$, ...
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1answer
59 views

Topology, Proof of function being continuous

Let $ (X_i,d_i),(Y_i,d_i^*)$, $i=1,\ldots,n $ be metric spaces. Let $ f_i:X_i \to Y_i, i=1,...,n $ be continuous functions. Let $$ X = \prod_{i=1}^{n} X_i , Y = \prod_{i=1}^{n} Y_i $$ and ...
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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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24 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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16 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
23 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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24 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
59 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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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
37 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
20 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
26 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
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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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1answer
31 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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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
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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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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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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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265 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
25 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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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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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
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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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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
21 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
121 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
68 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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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
39 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
35 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
90 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 $(0,..,0)$.