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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Show that function $f$ is not continuous in $x=0$ for all $c\in\mathbb{R}$

Show by $\varepsilon-\delta$-criterion that for each $c\in\mathbb{R}$, the function $f\colon\mathbb{R}\to\mathbb{R}$, $$ f(x)=\begin{cases}\frac{1}{x}, & x\neq 0\\c, & x=0\end{cases} $$ is not ...
2
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1answer
52 views

If $f \circ f$ continuous prove $f$ continuous

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ strictly increasing so that $f \circ f$ is continuous. Prove $f$ is continuous. I can prove this using sequences, but it's quite tedious. My question ...
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1answer
18 views

Global Lipschitz implies bounded in coefficient

Consider $g:\mathbb{R}^2\to \mathbb{R}$ of the form $g(x,y)=p(x)q(y).$ Assume $g$ is uniformly Lipschitz in $x,y$ in the sense that there exists $K>0$ such that for any $(x_1,y_1),(x_2,y_2)\in ...
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1answer
41 views

Show Lipschitz continuity of a function

I'm stuck trying to solve the following exercise: Let $f:\mathbb R^n \to \mathbb R^m$ a function with the property that, for all $v \in \mathbb R^n$, there is $L=L(v) > 0$ such that the ...
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10 views

Weak conitnuity of an operator [on hold]

What does it mean for an operator defined on an inner product space to be weakly continuous in its parameter?
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1answer
16 views

Continuity with norms defined via the supremum norm [on hold]

We look at the Vectorspace $$C_b^1(\Bbb R;\Bbb C):=\{f \in C^1(\Bbb R;\Bbb C):||f||_\infty \lt \infty, ||f'||_\infty \lt \infty \}$$ and define the Norms $||\cdot||_1$ and $||\cdot||_2$ through ...
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1answer
50 views

All Continuous function can be drawn? [duplicate]

I googled and came to know that there are many continuous functions which cannot be drawn by hand, like Cantor, Weierstrass functions etc. Now this question was asked in a college admission ...
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4answers
110 views

Where is $x^x$ continuous?

The idea of continuity of a function is something I come across quite regularly, but I've never really understood it well. I'm trying to fix that by looking at some interesting functions. What ...
1
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2answers
26 views

Number of discontinuous values

We have to find the number of values of $x$ at which the function $$ f(x) = \frac{2x^5-8x^2+11}{x^4+4x^3+8x^2+8x+4}$$ is discontinuous. I thought that since both numerator and denominator are ...
1
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2answers
60 views

Can continuity of real functions be “globally” characterized?

Most characterizations of pointwise continuous functions defined on an interval rely on "local" properties. That is, a function is continuous at $x_0 \in I$ if it satisfies some property ...
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1answer
31 views

Linear function: relation between linearity and continuity

Given a linear function $A$ between two normed Vectorspaces i have to show euquality of the follwing statements: $A$ is continuous There exists a point where $A$ is continuous $A$ is ...
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2answers
44 views

Prove or disprove continuity of two maps

Yet another time I need help to prove continuity of a certain map and don't know how to do it: Look at the vector space $$C_b^1(\mathbb R; \mathbb C) := \{f \in C^1(\mathbb R;\mathbb ...
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0answers
27 views

Equivalence of statements about a linear map

I need someone to help me solve the following exercise: Let $(X, ||\cdot||_X)$ and $(Y, ||\cdot||_Y)$ be normed vector spaces over a common field $\mathbb K$ $(\mathbb R$ or $\mathbb C)$. For a ...
3
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0answers
33 views

Is there a math notation/ term for “$f(x_n) \to 0$ iff $g(x_n) \to 0$”?

I have two real-valued functions $f,g$ defined over the $N$-dimension Real Euclidean space: $$ f,g: \mathbb{R}^N\to\mathbb{R}. $$ They satisfy this property: $$ \forall x_n \in \mathbb{R}^N: f(x_n)\to ...
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2answers
43 views

How do I examine f on continuity?

Let $f$ be defined as follows: $$f:\mathbb{R}^{2}\to\mathbb{R}:(x,y)\mapsto\begin{cases}\frac{xy^{2}}{x^{2}+y^{4}}&\text{if } (x,y)\neq (0,0)\\ 0&\text{if } (x,y)=(0,0)\end{cases}$$ How do I ...
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0answers
9 views

Checking of uniformly continuity of the following functions

Which of the following 4 functions are uniformly continuous? and which are not? I want to know the process/explanation of the solutions.
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1answer
27 views

Proof/disprove contunuity of a map [duplicate]

I need help with proving / disproving something: Look at the map $$\Phi: (C([0,1], \mathbb R), ||\cdot||_{\infty}) \to (\mathbb R, |\cdot|); \,\,\,\,\,\,\Phi(u) := \int_0^1 u²(t) dt$$ a) ...
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1answer
44 views

Is the given function $f$ continuous?

Problem Let $\mathbb{R}_l$ denote the reals with lower limit topology, and let $\mathbb{R}_l\times \mathbb{R}_l$ have the product topology. Then the map ...
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0answers
15 views

function nondecreasing in both variables, set of discontinuities is a nullset

Let $f\colon [0,1]^2\to\mathbb{R}$ be a function such that $g(x):=f(x,y)$ for any $y$ and $h(y):=f(x,y)$ for any $x$ are nondecreasing functions (the second variable is fixed). Prove that the set of ...
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0answers
21 views

(Lipschitz/Uniform) Continuity of a map [on hold]

I need help with proving / disproving something. I'm really bad at TeX so maybe someone can help me formatting. Look at the map $$Φ: (C([0,1]), \mathbb R), ||·||_{sup}) ~ \to ~(\mathbb R, |·|); ...
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1answer
34 views

a continuous function on $\mathbb{Q}$

Is there a continuous bijective function from $[0,1] \cap \mathbb{Q}$ to $\mathbb{R}$? I think that there is no such function. The set $|[0,1] \cap \mathbb{Q}|$ is countable and $|\mathbb{R}|$ is ...
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3answers
72 views

Is Inverse of a function continuous too?

I read an example from "Principles of Mathematical Analysis" by Rudin under the section 'Continuity and Compactness'. According to the example, Let $X$ be the half-open interval $[0,2\pi)$ on the ...
2
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1answer
41 views

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then its absolutely continuous?

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then is it true that $f$ is always absolutely continuous?
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1answer
31 views

What is an example of a uniformly continuous function but not absolutely continuous

Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?
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1answer
35 views

Geometric generation principle form constructing the Hilbert Curve

I have some questions on the generation of the Hilbert's space-filling curve. Any help to clarify doubts a-e would be very appreciated. The Hilbert's space-filling curve is a function ...
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1answer
46 views

Are eigenvalues (resp. unit eigenvectors) dependent continuously on elements $a_{ij}$ of a symmetric matrix $A$? [on hold]

Let $A(t)=(a_{ij}(t)),~(t\in \mathbb R)$ is a symmetric matrix such that $a_{ij}(t)=a_{ji}(t)$ is a real-valued continuous function. Let $\lambda_1(t) \ge \cdots \ge \lambda_n(t)$ is all of the ...
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0answers
22 views

Continuity of Holder functions

If a function taking values in $\mathbb{R}^n$ is $\alpha$-Holder continuous along lines parallel to the axes (uniformly on a compact set), is it continuous?
2
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2answers
75 views

Prove that $f(x)=\begin{cases} \frac{x}{x-4}, & x\not= 4 \\ 0, & x=4 \end{cases}$ is continuous.

Prove that the function $f(x)$ defined by $$ f(x)=\begin{cases} \dfrac{x}{x-4}, & x\not= 4 \\ 0, & x=4 \end{cases} $$ is continuous. My question is: Do I have to prove the two sides ...
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3answers
81 views

$\mathcal{f}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ restricted to sections is continuous implies continuity

Let $\mathcal{f}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that $\mathcal{f}$ restricted to each {$x=a$} is continuous and restricted to each section {$y=b$} is continuous and monotone.Prove that ...
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0answers
36 views

For a linear function the following are equivalent: continuity and Lipschitz continuity

Let $(X,||\cdot ||_X)$ and $(Y,||\cdot ||_Y)$ be normed Vectorspaces over a common field $\Bbb K$. Let $A:X \to Y$ be a linear function. I have to show that the following statements are equivalent: ...
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1answer
15 views

Continuity of Lipchitz constant of local lipschitz function

Suppose $f:\mathbb{R}\to \mathbb{R}$ be local lipschitz, which is equivalent to Lipschitz on compact sets. That is, for any $R>0$, there exists some $L >0$ such that $$\sup_{|x|,|y|\le ...
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3answers
46 views

Whether the function $f(x,y)$ is continuous at $(0,0)$

QUESTION: $$f(x,y)=\begin{cases}x \sin \frac{1}{y} + y \sin \frac{1}{x} & \text{if } xy \not = 0 \\ 0 & \text{if } xy = 0\end{cases}$$ Show that $f(x,y)$ is continuous at ...
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2answers
35 views

Infinite differentiability of a function with a removable discontinuity

How would I prove that $\frac x{e^x-1}$ is infinitely differentiable? (This question came up since the No 1 answer in Maclaurin series for $\frac{x}{e^x-1}$ states that the function is infinitely ...
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1answer
46 views

Show continuity or uniform continuity of $\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$

$\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | ); \: \: \: \: \: \: \phi(u):=\int_0^1 u^2(t) dt $ Is this function continuous or even uniformly continuous? (I know that the ...
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2answers
40 views

Proof that function on topological space is continuous if and only if 2 restrictions of it are

Topology such that function is continuous if and only if the restriction is. I've already seen this post but it didn't really help. The problem is the following: Let $X$ and $Y$ be topological ...
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0answers
11 views

Testing differentiability and continuity

Consider the following function $ f(x) = 0 $ if x is rational $ f(x) = x^2$ if x is irrational Then only one of the following statements is true which one is it ? a.) $f$ is differentiable at ...
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1answer
26 views

Lipschitz-continuity of $x\mapsto\frac{x}{||x||}$ in a general Banach space

Let $(X,||.||)$ be a Banach space. Assume we have constants $0<C_1<C_2<\infty$. Define the set $A:=\{x\in X\text{ }|\text{ } C_1\le ||x||\le C_2\}$. Is the map $f\colon A\rightarrow X$, ...
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0answers
60 views

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
5
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1answer
71 views

Differentiablity at $0$ of a function $f: \mathbb R \to \mathbb R$ which is twice differentiable in $\mathbb R \setminus \{0\}$

Let $f: \mathbb R \to \mathbb R$ be a function , twice differentiable in $\mathbb R \setminus \{0\}$ such that $f'(x)<0<f''(x) , \forall x <0$ and $f'(x)>0>f''(x) , \forall x >0$ ; ...
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1answer
60 views

Continuous or Differentiable but Nowhere Lipschitz Continuous Function

What is a real valued function that is continuous on a close interval but not Lipschitz continuous on any subinterval? What is a real valued function that is differentiable on a close interval but ...
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1answer
25 views

If a linear map $T:X^*\to X^*$ is norm-norm continuous, is it weak-star - weak-star continuous?

Let $X$ be a Banach space and suppose $T:X^*\to X^*$ is a linear mapping. If $T$ is norm-norm continuous, i.e. continuous from the normed space $X^*$ into the normed space $X^*$, is it also continuous ...
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1answer
24 views

How does this faulty system of integration change the nature of jump discontinuity?

Let's define a sort of faulty integral. For the purposes of this question we shall assume that this is the regular integral. This integral integrates all functions properly however it's gets confused ...
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0answers
11 views

deformation retraction as mapping cylinder

In Hatcher's Algebraic Topology, the mapping cylinder is defined as the quotient space of the disjoint union $(X\times I)\sqcup Y$ (where $I$ is the unit interval) of a continuous $f:X\to Y$, where ...
0
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1answer
44 views

Continuous function math question

This is the last question I have to answer for my math class. I thought I understood the concept of a continuous function, but I can't seem to get this one right. I only have 1 more submission ...
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2answers
16 views

Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism [closed]

Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism Requirements for a homeomorphism $f:X \rightarrow Y$: $f$ is ...
2
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3answers
23 views

$f$ continuous $\iff f(B(a,\delta))\subset B(f(a),\epsilon)$

My book says that when $f$ is continuous, we have that $\forall \epsilon>0$, there exists $\delta>0$ such that: $d(x,a)<\delta \implies d(f(x),f(a))<\epsilon$ Then, my book says that ...
0
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1answer
29 views

$\partial A$, when $A=\{x\in M: f(x)>0\}$ is the set $\{x\in M: f(x) = 0\}$

I have a question about the proof of this fact: $\partial A$, when $A=\{x\in M: f(x)>0\}$ is the set $\{x\in M: f(x) = 0\}$ The proof says the following: $$A = f^{-1}((0,+\infty))$$ Since ...
3
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3answers
83 views

$f \in C(\mathbb R)$ such that $f\Big(x+\dfrac 1n \Big) > f\Big(x-\dfrac 1n \Big) , \forall x \in \mathbb R , n \in \mathbb N$ ; is $f$ increasing?

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $f\Big(x+\dfrac 1n \Big) > f\Big(x-\dfrac 1n \Big) , \forall x \in \mathbb R , n \in \mathbb N$ , then is it true that $f$ is ...
0
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2answers
41 views

For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N

In order to prove: For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N I'm supposing that $x_n$ is convergent, that is: $$\forall \epsilon>0, ...
0
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1answer
14 views

How do I examine the function on continuity? How do I discuss and sketch the level lines of f?

How do I examine the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, $f (x, y) = (2x- y)\ \rm{sign}(4x-y)^2$ for continuity? How do I discuss and sketch the level lines of $f$?