Questions tagged [continuity]

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) or (general-topology)

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0
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1answer
42 views

For open sets $A,B \subset \mathbb R^n$ the map $x \to \lambda (A \cap (x + B))$ is continuous

I am working on the following: For $B \subset \mathbb R^n$, $x \in \mathbb R^n$ define $x + B := \{x + b | b \in B\}$ Let $A,B \subset \mathbb R^n$ be open sets with $\lambda(A) + \lambda(B) < \...
3
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1answer
63 views

Application of Rolle's

Suppose $q$ is a nonzero function of a real-variable such that $$u^2q''(u)+uq'(u)=u^2q(u)+q(u)$$ for all $u$. Assume there exist $x,y$ such that $q(x)=q(y)=0$. By Rolle's there exists $x<z<y$ ...
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Show that $\phi:(u,v)\in\mathcal I^2\mapsto f(u^{-1}\circ v)$ is uniformly continuous

Let $\mathcal I$ be the set of isometries of a vector space $(E,||.||_E)$ of finite dimension, $||.||$ be the norm associated to $||.||_E$ on $\mathcal L(E)$ and $f$ be a continuous function from $\...
3
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2answers
993 views

Let $f(x) = x^3-x^2–3x–1$ and $h(x)=\frac{f(x)}{g(x)}$ where $h(x)$ is a rational function such that

Let $f(x) = x^3-x^2–3x–1$ and $h(x)=\dfrac{f(x)}{g(x)}$ where $h(x)$ is a rational function such that (a) it is continuous every where except when $x=– 1$ (b) $\lim_{x\to\infty}=\infty$ (c) $\lim_{...
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0answers
31 views

Show that $O(n,\mathbb{R})$ the set of all orthogonal matrices is closed in $M(n,\mathbb{R})$

Show that $O(n,\mathbb{R})$ the set of all orthogonal matrices is closed in $M(n,\mathbb{R})$ Let $$f:M(n,\mathbb{R}) \mapsto \mathbb{R}$$ $$ f:A \mapsto det(A.A^t)$$ We know that $$ \text{det}(x_{i,...
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0answers
27 views

Constructing a continuous approximation for the Heaviside function

I have the following Heaviside function i.e: $$H(t)=\left\{\begin{array}{ll} 1 & \quad \mbox{if }\ t\geq 0, \\[0.1cm] 0 & \quad \mbox{if }\ t<0 . \end{array} \right.$$ Which is a ...
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0answers
21 views

Continuous function, Lebesgue measure

I'm trying to prove the next: For every Lebesgue measure set $A,B \subset \mathbb{R}^n$, $λ(A),λ(B)<\infty$ the map $x↦λ(A∩(B+x))$ is continuous, where $B+x={b+x:b∈B}$. I have already shown that ...
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2answers
145 views

Nowhere continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $f\bigl(f(x)\bigr) = \frac{f(2x)}{2}$ [closed]

I'm looking for a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at no point and satisfies the identity $$f\bigl(f(x)\bigr) = \frac{f(2x)}{2}$$ for all $x \in \mathbb{R}$. I was only able ...
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32 views

Continuity of two variable function $f( x,y)$ at $(0,0)$.

If function $f(x,y)=\begin{cases}\frac{sin((xy)^{2})}{(x^2+y^2)^{\alpha}(xy)^{\alpha}} & (x,y)\neq (0,0)\\ 0& (x,y)=(0,0) \end{cases}$ is continuous at $(0,0)$ , then $1$. $\alpha <1$. $2$....
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0answers
30 views

Continuity and differentiability of f(x,y) at the origin

Let $f$ be a function defined by $$f(x,y) = \begin{cases} \dfrac{\sin x - \sin y}{x-y} & x \neq y \\ \cos x & x = y \end{cases}$$ Study the continuity and differentiability of $f$ in the ...
5
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1answer
53 views

Theorem 10.2 Rudin

$\mathscr b(X)$ denotes the set of all complex-valued, continuous, bounded functions with domain $X$. I don't understand why is $L(h)$ equal of $\prod_{i=1}^k $ $\int_{a_i}^{b_i} h_i(x_i)dx_i$ and ...
2
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0answers
35 views

Baby Rudin 10.1

I want to be sure that I understand it correctly. As I understand it $\int_{I^k} f(x) dx $ = $\int_{a_{k-1}}^{b_{k-1}}$ ($\int_{a_k}^{b_k} f_k(x_1,...x_{k-1},x_k)dx_k$)$dx_{k-1}$ and to continue this ...
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1answer
34 views

How do I prove this statement about winding numbers and continuous maps?

I have the following problem: Let $D$ be a disk with boundary circle $C$ and let $f:D\rightarrow \mathbb{R}^2$ be a continuous map. Suppose $P\in \mathbb{R}^2\setminus f(C)$ and the winding number of ...
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1answer
21 views

Existence of a continuous function out of a metric space with specific properties

In a paper I am reading, the existence of a function as described below is claimed with no explanation. I can not figure out why such a function should exist. Can someone tell? Urysohn's lemma does ...
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2answers
38 views

Method for proving continuity for a complex function

We have that continuity for a complex function is defined as such: f is continuous at $z=z_0$ if it is defined in a neighborhood of $z_0$ and there exists a limit as: \begin{equation} \lim_{z\...
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1answer
29 views

I need the expression of a function which goes to infinity as x goes to 0 and has a finite integral

Is there a function similar to the one in the image that has a discontinuity in the origin, for which: $$\lim_{x \to \pm\infty} f(x)=0$$ and $$\lim_{x \to 0^{\pm}} f(x)=+\infty$$ and $$\int_{-\infty}^{...
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0answers
56 views

$f$ is continuous in $x$ using epsilon-delta definition

Show if $f$ is continuous in $x \iff$ for all $\epsilon>0$ there exists some $\delta >0$ such that for all $\zeta \in (a,b): |f(\zeta) - f(x)| < \epsilon$, whenever $|\zeta - x| < \delta$. ...
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0answers
53 views

Find all the functions satisfying $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$ for all real x,y [duplicate]

Find all the functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(x+y)=f(x)+f(y)$ and $f(x\cdot y)=f(x)\cdot f(y)$ for all $x,y\in\mathbb{R}$. Hint:You may need order properties and ...
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2answers
58 views

Nowhere continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = (f(x))^2$

Are there any nowhere continuous functions $f: \mathbb{R} \to \mathbb{R}$ satisfying the identity $f(f(x)) = (f(x))^2$ $\forall x \in \mathbb{R}$?
2
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1answer
48 views

Theorem 9.41 baby Rudin

The previous theorem is needed for $(9.41)$. I don't understand that how the sufficiently smallness of $h$ and $k$ guarantees that we have $|A-(D_{21}f)(x,y)|$ < $\epsilon$ for all $(x,y)$ $\in$ $...
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2answers
78 views

Is $\frac{x^2-1}{x-1}$ continuous at $x=1$? [duplicate]

A function $f$ is said to be continuous if $\lim\limits_{x \to a} f(x)=f(a)$ Now when we evaluate a function like $\frac{x^2-1}{x-1}$ as $x \to 1$ we find that the limit is 2. $$\lim\limits_{x \to 1}\...
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1answer
40 views

Does continuity of a map $f$ imply: $f^{-1}$ maps subbasis onto subbasis?

I am stuck with an exercise, maybe you can help me out here. Let $(X,\tau)$ be a topological space with subbasis $\mathcal{S}$ and let $f: X \longrightarrow X$ be a surjective map. Show that $f^{-1}(...
3
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1answer
38 views

Show that any two closed and bounded interval are homeomorphic in $\mathbb{R}$

Any two closed and bounded intervals are homeomorhpic in $\mathbb{R}$: If we want to show that the two sets $[a,b]$ and $[a_1,b_1]$ are homeomorphic we can consider the following map $f(x) \mapsto \...
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1answer
30 views

Proof that a nonzero point is continuous for g(x)

I have come across this problem in my math book for my numerical analysis class: Let $g(x)=\sqrt[3]{x}$. Prove that $g$ is continuous at a point $c \neq 0$. I start my proof off the typical way for ...
2
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3answers
1k views

Does band-limited imply continuous?

I was hoping someone could point me to an article or text which explores the connection between the continuity of a signal in the time domain and it being band-limited in frequency domain. (Update) ...
2
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3answers
2k views

Image of a connected space under a continuous map is connected, proof

In Munkres' Topology there is a theorem which states that the image of a connected space under a continuous map is connected. In the proof of this, they let $f: X \rightarrow Y$ be continuous, where $...
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1answer
25 views

Continuity of composition of root and floor function

Can someone give me a hint to prove the continuity of the following function: $f: \mathbf{R}\to\mathbf{R}, f(x):=\sqrt{\lfloor{x^2}\rfloor}$. I already proved the continuity of the root function and ...
0
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1answer
26 views

There is a set of continuous functions $f$ on [0, 1] with supremum metric (metric space). Proof that $\phi(f) = f(0) + f(1)$ is continous

Consider the set $C([0, 1]) = \{f : [0, 1] \to \mathbb{R} : f$ is continuous }, with the supremum metric $d_{sup}$ $$d_{sup}(f, g) = sup{|f(x) - g(x)| : x \in [0, 1]}$$ Let $\phi : C([0, 1]) \to \...
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2answers
48 views

Proving Lipschitz continuity (Real analysis)

A function $h : A → \mathbb{R}$ is Lipschitz continuous if $\exists K$ s.t. $$|h(x) - h(y)| \leq K \cdot |x - y|, \forall x, y \in A$$ Suppose that $I = [a, b]$ is a closed, bounded interval; and $g : ...
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1answer
48 views

Which of the following is definitely false?

The graph of the $f(x)$ function is given above. According to this graph, which of the following is definitely false? A) $\lim_{x\to -2}f(x)=1$ B) $\lim_{x\to 2^+}f(x)+\lim_{x\to 0}f(x)=4$ C) $f(x)$ ...
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2answers
31 views
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0answers
18 views

Prove that monotonic function defined on a measurable set is continuous almost everywhere

Theorem: If a function f defined on a measurable set D ⊂ R is continuous almost everywhere then f is measurable. Corollary: Monotonic functions defined on measurable sets are measurable. The ...
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0answers
28 views

Reference for convergent nets as continuous functions from $D\cup\{\infty\}$

If we have any directed set $(D,\le)$ then we can add a point $\infty\notin D$ and then consider the topology on $C(D)=D\cup\{\infty\}$ such that all points of $D$ are isolated and the local base at $\...
2
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1answer
171 views

Compact open topology on $\operatorname{GL}(n, \mathbb{R})$ coincides with Euclidean topology.

There are two ways to assign $\operatorname{GL}(n,\mathbb{R})$ topologies: as subspace of $\mathbb{R}^{n^2}$, or subspace of $\operatorname{Maps}(\mathbb{R}^n, \mathbb{R}^n)$ where the latter is ...
32
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1answer
781 views

Does there exist a space filling curve which sends every convex set to a convex set?

Does there exist a surjective continuous function $f:[0,1]\to [0,1]^2$ which maps every convex set to a convex set? Such a function could be considered an especially "regular" sort of space-...
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1answer
38 views

Property of periodic continuous mapping

Let $f\ne0\in\mathbb{R}^\mathbb{R}$ be a continuous mapping with period $p>0$. Is it true that $id\cdot f$ is not uniformly continuous? Let $a\in\mathbb{R}$ where $f(a)\ne0$ and $\delta>0$. If $...
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0answers
31 views

When does $f(x_i,y) \to f(x,y)$ follow from $f(x_i,y_i) \to f(x,y)$ and $y_i \to y$?

Say I have topological spaces $X$, $Y$ and $Z$, and a continuous map $f : X \times Y \to Z$. Say also that I have sequences $(x_i)$ in $X$ and $(y_i)$ in $Y$, along with points $x \in X$ and $y \in Y$ ...
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0answers
16 views

Showing parametrization of a screw-drive graph has a continuous inverse

By a screw-drive graph I mean the set $$S=\{(R\cos t,R\sin t, at):t\in(0,\infty)\}$$ The function $g(t):(0,\infty)\to S$, such that $g(t)=(R\cos t,R\sin t,at)$ is a continuous bijection. I'm trying to ...
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0answers
43 views

How do I prove that the following function $f$ is continuous?

Let $$f:\mathbb{R}^n \mapsto \mathbb{R}^n$$ $$f(x) = \begin{cases}x& \mbox{ if } ||x|| \le 1 \\ \frac{x}{||x||^2} & \mbox{ if } ||x|| \ge 1 \end{cases}$$ I am trying to use gluing lemma to ...
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0answers
73 views

Show that $p(x)$ has a root in the interval $(0,1)$. [duplicate]

Suppose that $a_{0}, a_{1}, a_{2}, ..., a_{n} \in \mathbb R$ satisfy $$\frac{a_{0}}{1} + \frac{a_{1}}{2} + \frac{a_{3}}{2} + ... + \frac{a_{n}}{n+1} = 0.$$ Show that $p(x) = a_{n}x^n + a_{n-1}x^{n-1} +...
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0answers
63 views

A function that is not continuous at any real number yet, is continuous at all real numbers. [closed]

This was a question I came across reviewing an old real analysis past paper. Give an example of a function f : R → R such that f is not continuous at any c ∈ R, but f is continuous at all c ∈ R. I ...
0
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1answer
29 views

f(x,y) not Differentiable on (0,0)

Suppose that $f: \mathbb{R}^2\to \mathbb{R}$ is constructed to be continuous s.t. i) $f(x,y)= 0$ unless $x>0$ and $x^2<y<3x^2$ ii) for each $x>0$, $f(x,2x^2)=x$ iii) $0 \leq f(x,y) \leq x$,...
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0answers
25 views

Want to find a proof of the Extension Theorem

Question: I am trying to find a proof of the following extension theorem: If $E$ is a subset of $X$ where $E$ is closed, and $f:E\to\mathbb{R}$ is continuous, then there exists $F:X\to\mathbb{R}$, ...
1
vote
2answers
95 views

How to show that $x^{1/n}$ is uniformly continuous

Let $n\in\mathbb{N}$ and $n\geq 2$. For $x,y\in\mathbb{R}_{\geq 0}$, we have $$(x-y)=(x^{1/n}-y^{1/n})\sum_{i=0}^{n-1}(x^{1/n})^{n-i-1}(y^{1/n})^{i},$$ which can be equivalently stated as $$\frac{x-y}{...
4
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2answers
11k views

Is it possible to have a function differentiable but not continuous in a given interval?

Is there any possible function that is not continuous but differentiable in a given interval. It sounds non-logical to me since differentiation is a special limit function in itself therefore non-...
2
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1answer
39 views

Consider $Y = \mathbb{N} \cup \{ \infty\}$ and $X = \{ \frac{1}{n}\mid n \in \Bbb N\} \cup \{0\}$

Consider $Y = \mathbb{N} \cup \{ \infty\}$ and $X = \{\frac{1}{n}\mid n \in \Bbb N\} \cup \{0\}$ Consider $X$ with usual metric and let us consider a bijective map $f:Y \mapsto X$ where $f(n) = \frac{...
1
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2answers
87 views

Discontinuous functionals on infinite dimensional topological vector space [closed]

It is easy to see that every infinite dimensional normed vector space has discontinuous functionals. My question is: Is this also true for general topological vector spaces?
0
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1answer
51 views

Finding distance between 2 functions, given their inner product

Let $u=f$, $v=g$ be continuous functions on $[0,1]$ where $f(x)=x^2+x$ and $g(x)=x+1$ with inner product $\langle f,g\rangle =\int_0^1f(x)g(x)dx$. Find the distance between u and v. We have only just ...
2
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0answers
32 views

Possible extension by continuity of the function $f(x,y)=\frac{ye^{x^2}-x\sin(y^2)-y}{x(x^2+y^2)}$

Find the domain and its limit points where the function $f:\Bbb R^2\to\Bbb R$ is given by $$f(x,y)=\frac{ye^{x^2}-x\sin(y^2)-y}{x(x^2+y^2)}$$ can be extened by continuity. My attempt: I rewrote the ...
0
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0answers
16 views

Sketch check, IVP and continuity

Exercise : A function $f$ is increasing on $A$ if $f(x)\leq f(y)$ for all $x<y$ in A. Show that if $f$ is increasing on $[a,b]$ and satisfies the intermediate value property, then $f$ is continuous ...

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