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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Is a function continuous iff its restriction to each element of an open cover is continuous

Let $(X;T_1)$ and $(Y;T_2)$ be topological spaces and let $A$ and $B$ be nonempty subsets of $X$ with $A\cup B= X$ Suppose $f:X\rightarrow Y$ is a function. Then prove or disprove: (a) if $f_A$ and ...
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25 views

Is the Inverse of the Vectorised Solid Angle Equation for $n$ Circular Discs Continuous?

I have a continuous function$^{*1}$ that takes in 3 arguments, and returns 24 outputs. I want to know if the inverse of this function is continuous. The 3 input arguments are the x, y, and z position ...
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Proving $\lim_{x\to c} g\circ f(x)= g(b)$ without sequential criterion

Pardon my English beforehand. I want to prove, without using the sequential criterion for continuity, the next theorem: Let $f$,$g$ be defined on $\mathbb R$ and let $c\in \mathbb R$. Suppose ...
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Does right continuity imply only countably many discontinuities? [duplicate]

Does right continuity imply only countably many discontinuities? That is, if $f:\mathbb{R}\rightarrow \mathbb{R}$ is right continuous then does it only have countable many discontinuities? Thanks
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How would I finish this continuity proof?

I have a multivariable function $f$ with $$f(x, y) = \begin{cases} \frac{x^2+y^2}{y} & \text{if }y \neq 0\\ 0 & \text{if }y = 0 \end{cases}$$ and want to show that it is continuous at $(0, ...
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How to prove that $f(x,y)=3+2x+y$ is continuous?

The question is to prove that the function $f(x,y,z) = 3+2x+y$ is continuous everywhere. My approach uses the delta-epsilon method. $|(x,y)-(a,b)|\lt \delta$ then $|f(x,y)-f(a,b)|$. All I did was ...
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Can monsters of real analysis be tamed in this way?

Consider the Weierstrass Function (somewhat generalized for arbitrary wavelengths $\,\lambda > 0$ ): $$ W(x) = \sum_{n=1}^\infty \frac{\sin\left(n^2\,2\pi/\lambda\,x\right)}{n^2} $$ $W(x)$ is an ...
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25 views

Does weakly differentiable and $L^{\infty}$ imply continuity

Suppose $\Omega \subset \mathbb{R}^d$ is open, connected and bounded. Is $$W^{1,1}(\Omega)\cap L^{\infty}(\Omega) \subset C(\bar{\Omega})?$$ Here $W^{1,1}$ denotes the space of all weakly ...
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Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
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38 views

Uniform continuity of $\arctan x$

Check if $\arctan x$ is uniformly continuous on $\mathbb R$ If I'll show that it's contious on $[0,\pi/2]$ then because it's periodic it would be continuous on $\mathbb R$. So by the definition ...
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Question about limits and Mean Value Theorem

Let $f:(a,b) \rightarrow \mathbb{R}$ and $g:(a,b) \rightarrow \mathbb{R}$ be differentiable on (a,b) with $g'(x) \neq 0$ for all $x$ in $(a,b)$. Suppose $\lim_{x \to b-}\dfrac{f'(x)}{g'(x)}$ ...
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42 views

Is function continuous at 0?

Is $f:[0, \infty) \rightarrow \mathbb{R}$ $f(x)=[x^{1/2}]$ continuous at 0? My Attempt Now using the limit method, and as the function is only defined on $[0, \infty)$ $$\lim_{x \to +0}=\lim_{x ...
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12 views

About maximum function and continuity

Let $\bar{x}\in\mathbb{R}^n$, $R>0$, and $P$ metric space. If $f:\bar{B}(\bar{x},R)\times P\rightarrow\mathbb{R}$ is a continuous function. We define $F:P\rightarrow\mathbb{R}$ by ...
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49 views

Show that $f*(x) = \sup \{ f(y) : a \leq y \leq x \}$ is a non-decreasing continuous function

I am currently working on a problem and stuck on it. Here is the problem (it comes form Elementary analysis, the theory of Calculus by K. Ross P.153): Q: Let $f$ be a continuous function on [a,b]. ...
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Let $f: X \rightarrow [0,1]$ where $f^{-1}([0,a))$ and $f^{-1}((b,1])$ are open sets in $X$ prove $f$ is continuous

Let $f: X \rightarrow [0,1]$ where $f^{-1}([0,a))$ and $f^{-1}((b,1])$ are open sets in $X$ for each $0<a,b<1$ prove $f$ is continuous The problem with question it's not clear which topology ...
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23 views

Equivalence of different definitions of continuity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real function. $f$ is continuous at point $c$ iff $$(\forall\epsilon>0)(\exists\delta>0)(\forall ...
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How to show that the following function is not continuous?

Let $f: \mathbb R \rightarrow \{a,b\}$ for some $a,b \in \mathbb R$ such that $a \neq b$. I claim that such a function is not continuous on $\mathbb R$ since there exists a point $c \in \mathbb R$ ...
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$ \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin(t^2)\; dt = \frac{(-1)^n}{c}, \text{ where } \sqrt{n\pi} \leq c \leq \sqrt{(n+1)\pi}. $

The following is a problem from Apostol Vol 1 Calculus from the section: Continuity. Since Differentiation hasn't been introduced yet, the objective is to solve it without direct reference to ...
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Prove there exists a point $c$ such thst $f(c)=c$ for the following function

If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function with $f(0)=2$ and $|f'(x)| \leq 1/2$ for all $x$ then there is a point $c$ such that $f(c)=c$ . My Attempt Let ...
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Functions continuity

I have a question regarding continuity of a function that has 2 parts and 2 variables: $$f(x) = \begin{cases} \dfrac{\arctan x}{1 + x^2}, & \text{for $x\ge $ 0} \\ A e^x + B, & \text{for ...
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Does there exist a continuously differentiable function with the following properties?

Does there exist a continuously differentiable function $f: [1,5] \rightarrow \mathbb{R}$, such that $f(1) \lt 0, f(5) \gt 3$ and $f'(x) \leq e^{-f(x)}$? Now do I just integrate it to get $f(x) ...
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Prove that f is differentiable at $0$! Not continuous though, Right!?

Suppose $f(x)$ equals $x^2$ when $x\in \mathbb{Q}$ and $0$ when $x \not\in \mathbb{Q}$ Prove that $f$ is differentiable at $0$ and find the derivative $f'(0)$ Shouldn't this be obvious, since $x^2$ ...
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A basic question about upper hemicontinuity

Given a correspondence $f:X\rightarrow 2^X$, suppose X is a closed simplex in $\mathbb{R}^n$, and $f$ is compact-valued. We say $f$ is upper hemicontinuous if, $\forall x\in X$ and every open subset ...
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Is $f(x)=\log(1+x^2)$ uniformly continuous on $(0,\infty)$?

Is $f(x) = \log(1+x^2)$ uniformly continuous on $(0,\infty)$? My work: Looking at the graph and knowing that $\log$ considered a "slow-growing" function, my guess is that $f(x)$ is uniformly ...
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33 views

Continuity of a function in the product topoogy

Hi everyone I would like to understand if my reasoning is correct. Let $X$ be the space of sequences with values in the interval $[0,1]$, i.e. if $\mathbb{N}$ is the set of natural numbers, $x\in X$ ...
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Example of continuous function without fixed point.

I need to find an example of a continuous function without a fixed point, and this is what I've come up with: As {1} is not in the (co)domain, I can evade all $x$ for which $f(x)=x$ up until I ...
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integral of a product of functions being $0$

Suppose we have a continuous function $f$ on $[a,b]$ such that for all integrable functions $g$ such that $\int_{[a,b]}g=0$, $\int_{[a,b]}fg=0 $. Show that $f$ must be constant. Well, it's clear ...
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Checking if a piecewise defined function in two variables is continuous

How would I check if the following function is continuous? $$ f(x, y) = \left\{ \begin{array}{ll} \sqrt{1 - x^2 - y^2} & \text{, if } x^2 + y^2 \leq 1\\ 0 & \text{, otherwise} ...
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A question about differentiable functions and step/jump discontinuities

I got this question: Let $f$ be a differentiable function defined on an interval $I$, Must it be the case that $f'$ (the derivative of $f$) doesn't have step/ jump discontinuities on the interval $I$ ...
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Continuity of Shannon Entropy with respect to KL-Divergence distance

I am trying to prove the following statement which seems to be trivial but I cannot: Suppose $p$ and $q$ are two distributions. If $D(p||q) < \epsilon$ and $D(q||p) < \epsilon$ for some ...
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$g(x)$ is continuous on $\mathbb{R}$ st $g(x)=g(x^2)$. Prove that $g(x)$ is constant.

For $x>0$, $g(\sqrt{x})=g(x)$ and similarly $g(x^{\frac{1}{2^n}})=g(\sqrt{x})=g(x)$ for $n \in \mathbb{N}$ Thus taking $\lim_{n\to \infty}$ both sides we get g(1)=g(x) $\forall x>0$ and ...
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Measure theory integration question involving continuous function

Quick measure theory question. Given that $\Omega \subset \mathbb{R}^{n}$ and $f$ is continuous on $\Omega$. How would you show that if $$\int_{\Omega}f \, dx = 0$$ Then $f = 0$ everywhere? Thanks ...
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Prove the following statements

Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous with $f(0)=0$ and $f(1)=1$. For the following you may apply standard results without proof provided you state them carefully; $(1)$ If ...
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222 views

Find a function that satisfies the condition.

Let $\epsilon > 0$ be fixed and $t$ a variable that takes values in the universal covering space of ${\mathbb{C} \setminus \{0\}}$. Find a continuous function $f(s$) such that $$|t \log t| = |t| ...
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Differentiating Dynkin's Formula

Let us assume that we are given a right-continuous, non-explosive continuous time Markov chain $(X_t)_{t\geq0}$ with infinite state space in $\mathbb{N}_0^d$. (Think, for instance, about a Markov ...
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26 views

how to find the coefficient for a function to be continous at all $x$

I'm having a problem solving this question, we have just learnt it at school today and this is my homework. Could you help me please? Find the values of a such that $f$ is continous for all values of ...
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37 views

Problem of continuous function

Define the function $g(x) = x^2\cos\frac1x$ for $x\ne 0$. What should be the value of $g(0)$ if $g(x)$ is a continuous function? Explain your work and justify your answer. Frankly, I have no ...
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Does there exist an unbounded function that is uniformly continuous?

I know that $1/x$ is unbounded on $(0,5)$ (for example) and that since it is unbounded, it is not uniformly continuous. Does a function have to be bounded to be uniformly continuous? I don't think ...
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When the inverse image of an *open* set is *closed*

Let $X$ and $Y$ be topological spaces. Assume that $f\colon X\to Y$ satisfies that the inverse image of any open set in $Y$ is closed in $X$ (as opposed to the definition of continuity). Can anything ...
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How to check if this function is semicontinuous

Could you tell me how to check that this functions are semicontinuous? $(X, \tau)$ - topological space, $ \ X \neq \emptyset$, $ \ f: X \rightarrow \bar{\mathbb{R}}$, $ \ \bar{\mathbb{R}} = [- ...
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If $g$ is continuous and $f$ is s.t $f=g$ for $|x|<1$ then $f$ is continuous at 0

If $g:\mathbb{R} \rightarrow \mathbb{R}$ and $f:\mathbb{R} \rightarrow \mathbb{R}$ is such that $f(x)=g(x)$ for all $|x|<1$ then $f$ is continuous at 0. Attempt; My claim is this statement ...
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Can someone explain the concept of continuity and differentiability for functions of several variables?

Can someone explain the concept of continuity and differentiability for functions of several variables? Illustrated examples will definitely help, on how to solve problems(or establish proofs) of the ...
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25 views

Show continuity and holomorphism for a function

Let $A = \left\{ z \in D_r \big| \; Im(z) \geq 0 \right\}, f : A \rightarrow \mathbb{C}$ continuos on $A$, holomorphic on the inner of $A$ and real-valued for $\left]-r,r\right[$. Let further be ...
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Show that: $\exists x \in \mathbb{R}. \left|P(x)\right| = e^x$

Show that: $\exists x \in \mathbb{R}. \left|P(x)\right| = e^x$. where $P(x)$ is a polynomial different from the zero-polynomial. Obviously, for every $y \in (0, \infty)$ there's $x$ such that $e^x ...
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34 views

Topological Rings and Homothecy

I am trying to show that in a topological ring $A$, if left homothecy $x \mapsto ax$ is continuous at $x=0$ for all $a \in A$ and multiplication $\mu$ is continuous at $(0,0)$, then multiplication is ...
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58 views

Continuity and Differentiation on a interval

$$f(x) = \begin{cases} x\sin(1/x), & \text{if $x$ $\ne$ $0$} \\ 0, & \text{if $x$ = $0$} \\ \end{cases}$$ Is $f$ continuous on $(-1/\pi$, 1/$\pi$)? Is $f$ differentiable on $(-1/\pi$, ...
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33 views

Is $[0,1]^{[0,1]}$ Hausdorff and first-countable?

I'm trying to determine if $[0,1]^{[0,1]}$ is Hausdorff or first-countable. What I know until now, is that $[0,1]^{[0,1]}$ has the product topology, then if $x\in [0,1]$ and $U$ open in $[0,1]$ the ...
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42 views

What does it means that sequences characterize closed sets and functions?

A text book I'm reading says at one point the following: "In metric spaces are sequences the ones which chacterize closed sets and continuous functions". What is exactly the meaning of that ...
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65 views

If $f,g$ are uniformly continuous prove $f+g,fg$ are uniformly continuous

Suppose $f:E \rightarrow \mathbb{R}$ and $g:E \rightarrow \mathbb{R}$ are uniformly continuous, where $E$ is a subset of $\mathbb{R}$. Show that $f+g \ \ and \ \ fg$ are uniformly contiuous, what ...
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35 views

Is it continuous at $(0,0)$?

$$f(x,y)=\begin{cases} \frac{xy}{x^2+y^2}, \text{ if } x^2+y^2\neq 0 \\ 0, \text{ if } x^2+y^2=0 \end{cases}$$ Is it continuous at $(0,0)$?