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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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 ...
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1answer
18 views

Showing Intermediate Value property and closed preimage implies continuity

Let $f : [0,1] \to \mathbb{R}$ be a function satisfying the Intermediate Value property. Assume that for any $y \in \mathbb{R},$ the preimage $f^{-1}(\{y\})$ is closed. Prove $f$ is continuous. ...
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34 views

Prove the function is continious.

If the function $f(x)$ is continious at $x=0$, using definitions show that $f(rx)$ is continious at $x=0$. Here $r$ is a real number.
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1answer
18 views

Error term between $f(x)$, its average value and value at midpoint

Let $f$ be a smooth function on interval $[a,b]$. Define the average $\bar{f}=\dfrac{1}{b-a}\int_a^bf(y)\,dy$ and $\bar{x}=\dfrac{a+b}{2}$, then for any $x\in [a,b]$, we can write ...
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2answers
70 views

Can we define a metric on $Y$ such that all continuous mappings $f:X\rightarrow Y$ are constant?

Given that $Y$ contains more than one element and let $X$ be the real line equipped with the standard metric. Then can we define a metric $\sigma$ on $Y$ such that every continuous mapping ...
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1answer
28 views

continuous random variable expectation and variance [on hold]

you have a continuous random variable X uniformly distributed, and E(X) = 3, calculate the V(X) am stuck, how am i supposed to get the variance with no function in the first place
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1answer
24 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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0answers
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Is the mapping “positive stochastic matrix onto its Perron-projection” continuous?

I am dealing with a topological question concerning the mapping that maps a positive stochastic matrix onto its invariant distribution. I am asking myself if such a mapping is continuous (or ...
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1answer
43 views

Continuous in topology [on hold]

Let $f$ be a continuous function from the closed unit interval $[0,1]$ to itself. Show that there exists $t∈[0,1]$ such that $f(t)=t$. P.S Not use intermediate value theorem. Tomorrow this problem ...
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2answers
36 views

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 ...
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1answer
57 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 ...
2
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1answer
46 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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0answers
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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4answers
112 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 ...
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1answer
18 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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2answers
29 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 ...
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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
32 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
45 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
34 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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0answers
38 views

Riemann integral and continuity

I have a Riemann integrable function on compact set [closed interval in reals]. I want to apply Stone-Weiestrass theorem from Rudin's. I need continuity of Riemann integrable function and I don't know ...
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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
10 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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5answers
251 views

A-noncompact, Does there **always** exist a continuous function $f: A \to \mathbb R$ which is bounded but does not assume extreme values?

It's well known that if $ A \subset \mathbb R$ is compact then every continuous function $f:A \to \mathbb R$ is bounded and assume extreme values .So the obvious question is: Given any non compact ...
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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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22 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
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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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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37 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
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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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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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 ...
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1answer
36 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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2answers
41 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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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 ...
2
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1answer
42 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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23 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?
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2answers
76 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
82 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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1answer
62 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
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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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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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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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 ...
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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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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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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
41 views

Can you give me an example of a function that is either upper OR lower quasi-continuous but not both?

A function $f: X \rightarrow \mathbb{R}$ is said to be upper (lower) quasi-continuous at $x \in X$ if for each $\epsilon >0$ and for each neighbourhood $U$ of $x$ there is a non-empty open set $G ...