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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Proof of Lipschitz continuity of coordinate map in $C[0,\infty)$

consider continuous function space $C[0,\infty)$ with metric $$d(\omega_1,\omega_2)=\sum_{n=1}^\infty\frac{1}{2^n}\left[(\sup_{t\in[0,n]}|\omega_1(t)-\omega_2(t)|)\wedge 1\right]$$ where ...
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1answer
41 views

Is a function continuous if sequence definition holds only over 'almost all sequences'?

By the definition of a continuous function, $f:\mathbb{R}\to\mathbb{R}$ is continuous at $x$ if and only if for any sequence $x_n\to x$, it also holds that $f(x_n)\to f(x)$. My question is as ...
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64 views

Prove that if $f(x_n) \rightarrow f(x)$ for every continuous real-valued function in the metric space M, $x_n \rightarrow x$ on M.

The problem goes like: Suppose that we are given a point $x$ and a sequence $x_n$ in a metric space M, and suppose that $f(x_n) \rightarrow f(x)$ for every continuous real-valued function $f$ on M. ...
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38 views

If $f$ is continuous and $A$ is path-connected, then $f^{-1}(A)$ path-connected [closed]

If $f:\mathbb{R^n}\rightarrow\mathbb{R^m}$ is continuous and $A\subset\mathbb{R^m}$ is path-connected, then $f^{-1}(A)$ path-connected?
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65 views

The equivalence of open-set definition of continuous mappings to the $\epsilon-\delta$ definition

A function $f: M\rightarrow N$ is defined to be continuous iff $\forall$ open set $U\subseteq N$, the preimage of $U$ is also open. I was trying to prove that such a definition would be equivalent to ...
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2answers
61 views

Is $f(x)=\frac{|x|^2}{x}$ continuous?

$$f(x) = \begin{cases} \frac{|x|^2}{x}, & \text{if $x \neq 0$} \\ 0, & \text{if $x=0$} \end{cases}$$ Can someone please explain if f is continuous? Assume $x$ is a complex number Hints ...
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31 views

Proof of a continuity for a function represented by an integral.

Please think this problem easy. I faced the following problem the other day. Let $f\in C(0,1]\cap L^{1}(0,1)$. Prove that the function $$ t\mapsto\int_{0}^{t}\frac{f(\tau)}{\sqrt{t-\tau}}d\tau ...
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2answers
37 views

How do I determine $a, b $ for which $\lim\frac{\sqrt{x²+ax}+b}{x²-1}=\frac{1}{2}$ when $x$ go to $-1$does satisfied?

How do i determine $a$ and $b$ for which : $$\lim_{x \to -1}\frac{\sqrt{x²+ax}+b}{x²-1}=\frac{1}{2}$$ does satisfied ? Attempt : I have used l'hopitale rule and substitution i have got $a=0$ and ...
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0answers
51 views

Prove there is a unique p such that the integral holds true

I am looking at the following problem: $[a,b]$ a closed interval in $R$ and $A$ and $K$ are continuous real-valued functions on $[a,b]$ and {$(x,y) \in E^2$: $a \leq y \leq x \leq b$} respectively. ...
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1answer
29 views

Continuity and Differentiability Q

We have f = e^(-1/|x|) if x is not equal to 0 and f(0) = p. Question 1: for what value(s) of p is f differentiable at 0? Question 2: is f' continuous for the values found in question 1? What I tried ...
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384 views

Creating an increasing function $g$ given a continuous function $f$, such that $|f(x)-f(y)| \leq |g(x)-g(y)|$

Given a continuous function $f$ over an interval, must there exist a continuous, increasing function $g$ such that for all $x,y$ $$|f(x)-f(y)| \leq |g(x)-g(y)|$$ I've tried assuming the opposite, ...
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1answer
42 views

Unbounded continuous function, on a set with a convergent sequence [closed]

Say $S \subseteq \mathbb{R}$ and we suppose there exists a sequence $(x_n)$ in $S$ converging to a number $x_0 \in S$. Is there an unbounded continuous function on $S$? (real analysis, from Ken Ross' ...
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2answers
81 views

A continuous function on a closed subset of real line can be continuously extended [duplicate]

I need help with this question as I'm not sure Let $f$ be a continuous function on $K$ which is closed in the real line. Show that there exists a continuous $F$ on all of $R$ such that $F(x)=f(x)$ on ...
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2answers
28 views

f: M $\rightarrow $ R is continuous if and only if for every a $\in$ R, $X_a$ and $Y_a$ are open sets

$f: M \rightarrow \mathbb{R}$ is continuous if and only if for every $a \in \mathbb{R}$, $X_a$ = {x $\in$ M | f(x) < a} and $Y_a$ = {x $\in$ M | f(x) > a} are open sets. ( $\implies$) If f is ...
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38 views

Continuity of a two variables function

I would like to study the continuity on $\mathbb{R}^2$ of these functions : $ 1) f(x,y)=\frac{y}{x^2}e^{-\frac{|y|}{x^2}}$ if $x \neq 0$ $0$ if $x=0$ $ 2) f(x,y)=(x^2+y^2)\sin(\frac{1}{xy})$ if ...
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0answers
48 views

Between uniform and pointwise convergence

I know that uniform convergence of a sequence of continuous functions $f_n$ on some set $S$ implies continuity of the limit function $f$ on that set. I also know that pointwise convergence of the ...
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1answer
37 views

Function continuity $(x^2 - 1)/( x - 1)$

What can you say about the function continuity of $$\frac{x^2 - 1}{x - 1}$$ at $x = 1$? There is an asymptote at $x = 1$, however the limit as $x$ goes to $1$ becomes $2$ because $f(x) = x + 1$ after ...
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2answers
45 views

Continuity of the function $\mathbb{R}^k \to\mathbb{R}: x\mapsto \ln(1+ \lVert x \rVert)$ [closed]

Examine the continuity of the function $f\colon\mathbb{R}^k \to \mathbb{R}$ defined by $f(x) = \ln (1+ \lVert x \rVert)$, where $\lVert\cdot\rVert$ is a norm.
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46 views

Is Lipschitz “type” function Continuous?

Suppose $f:\mathbb R \to \mathbb R$ is a function satisfying $f(x)-f(y) \leq k \vert x-y \vert$ for some $k \in \mathbb R$ and $\forall x,y \in \mathbb R$. Is $f$ continuous and differentiable? By ...
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1answer
29 views

Show that $T$ is not a homeomorphism

Let $X$ be the space of all polynomials in one variable over $\mathbb R$ .If $p=a_0+a_1x+\cdot\cdot\cdot\cdot +a_nx^n$ Define :$||p||=|a_0|+|a_1|+\cdot\cdot\cdot\cdot+|a_n|$. Prove that $T:X\to ...
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158 views

Let $f(x) = 1$ for rational numbers $x$, and $f(x)=0$ for irrational numbers. Show $f$ is discontinuous at every $x$ in $\mathbb{R}$

I am working on this proof, and wanted someone to check it and to help me understand what is happening in case (ii). The proof: Let $f(x) = 1$ for rational numbers $x$, and $f(x)=0$ for irrational ...
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35 views

Find the number a that makes $f(x)$ continuous everywhere?

$$ f(x)= \begin{cases} ax^2-3 & \text{if $x\leqslant 2$,} \\ 2ax+3 & \text{if $x > 2$.} \end{cases} $$ $$\lim_{x\to 2^-} ax^2-3 = 4a-3$$ $$\lim_{x\to 2^+} 2ax+3 =4a+3$$ $$ ...
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15 views

Functions that change definition with the type of input

Here are some functions that I came over in a question in my mathematics book of chapter continuity. Note that $I$ is the irrational numbers in the following definitions. $$f(x)=\begin{cases} 1 ...
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28 views

Weierstrass continuity vs sequential continuity

The real function $f(x)$ defined so that $f(x)=x$ when $x \in \mathbb{Q}$ and $f(x)=1 $ when $x \notin \mathbb{Q}$ is Weierstrass continuous. But, it doesn't have sequential continuity. If you try ...
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How discontinuous can the limit function be?

While I was reading an article on Wikipedia which deals with pointwise convergence of a sequence of functions I asked myself how bad can the limit function be? When I say bad I mean how discontinuous ...
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Calc 1 question regarding uniformly continuous

Let $f$ be an uniformly continuous function on $ \mathbb{R} $ with $ f(\sqrt{n}) = 0 \ \ \forall n \in \mathbb{N} $ Does this imply: $\lim\limits_{x \to \infty} f(x) = 0 $ ?
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How to prove that $x^{\frac{1}{n}}$ is continuous?

If there is a function $f$ such that:- $$ f:\mathbb{R}^{+}\rightarrow\mathbb{R}\\ f(x) = x^{\frac{1}{n}}, n\in \mathbb{Z}^{+} $$ How can we prove the continuity of above function using the ...
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1answer
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Prove that a group law of the Heisenberg group is continuous.

The group law on $\mathbb{H}^n$ -Heisenberg group- is given as follows: $(s,x,y)·(s′,x′,y′)=(s+s′+ ω(x,y;x′,y′),x+x′,y+y′)$, how can I prove that this group law is continuous? Many thanks.
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Prove that $f:V \times V\to V$ $f(v,w)=v+w$ is continuous

Let $V$ be a vector space with a metric $d$.Prove that if $d(v+w,w+z)=d(v,w)$ for all $v,w,z \in V$ then $f:V \times V\to V$ $f(v,w)=v+w$ is continuous Let $\epsilon>0$ and let $(v_0,w_0)\in ...
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Determine whether a function can be extended

Given that $x^2+y^2 < 1$ I have $$\lim_{(x,y)\rightarrow(1,0)} \frac{y}{ \sqrt{1-x^2-y^2}}$$ $$\lim_{(x,y)\rightarrow(1,0)} \frac{y^m}{ \sqrt{1-x^2-y^2}}$$ where $m > 1$. I'm supposed to ...
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2answers
26 views

Continuity of a multivariable function

I have this function: $$f(x,y)=\begin{Bmatrix} \frac{x^2+y^2}{y^2} &if (x,y)\neq 0) \\0 &if (x,y)=0) \end{Bmatrix}$$ I am asked to evaluate the continuity. I replaced 'y' by 'mx', then i ...
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A specific request on a real analysis book

I would like to find a book on real analysis having the following properties (or different books-references for that matter): Treating metric spaces and completeness/compactness/connectedness. Here ...
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3answers
64 views

Is the endpoint of a domain automatically an extreme point?

Consider the function $$f(x)=\begin{cases} \sin(1/x) &\mbox{if } x>0 \\ 0 &\mbox{if } x=0.\end{cases}$$ Does this function have a minimum at $f(0)$ ? I did Google the question, and ...
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26 views

continuity of brownian motion

Let $s \in ]0,1[$,$\epsilon>0$ with $(1+\epsilon)^2(1-s)>1+s$; $h(t)=\sqrt {2t\log(1/t)}$. Define $$K_n=\{(i,j) \in N^2:0 \leq i<j<2^n\text{ and }j-i \leq 2^{ns} \}\\ ...
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1answer
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Is it always possible to simultaneously slice two rain drops each exactly in half with a single straight-line cut?

In theory, can you always simultaneously slice two rain drops on the windscreen each exactly in half with a single straight-line cut, no matter the shapes of the rain drops nor their location on the ...
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32 views

Evaluate the continuity of this function

Ok so after countless hours looking at video about epsilon-delta, I still can't understand it. I am asked to evaluate the continuity of this function using epsilon-delta. $$f(x,y) = \begin{Bmatrix} ...
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2answers
33 views

Continuity on a Set

Suppose f is continuous on a S $\subset$ R. Show that the set $$D = \{ x ∈S: f(x) = 0\} $$ is closed I'm having trouble proving this. I tried the epsilon-delta definition of continuity: Since, f is ...
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26 views

Multivariable continuity in ball

How do you show continuity for ball-based functions such as $$f:B[(0,0),1)]\rightarrow\mathbb{R}, \space f(x,y) = \sqrt{(1-(x^2+y^2)}$$
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Extending functions to homeomorphism

Is there a homeomorphism $f:(0,1) \to \mathbb{R}$ such that $f$ (co)restricts to a homeomorphism $f:(0,1)\cap \mathbb{Q} \to \mathbb{Q}$? I am a bit rusty in general topology, but I think that ...
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1answer
30 views

Delta-Epsilon Continuity of a Dirichlet-like function

I'm trying to determine whether the function $$f(x) = \begin{cases} x, &\text{ if $x$ is rational}, \\ 1, &\text{ otherwise}. \end{cases}$$ is continuous (everywhere). I believe that it's ...
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1answer
26 views

$|x^{-1}-y^{-1}|=|x-y|/|x||y|$ in a normed ring

I hit a slight snag when trying to prove that the inverse function $x\mapsto x^{-1}$ on the unit group is continuous in a ring with an absolute value, so I'd like some confirmation that the theorem is ...
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2answers
102 views

Prove that $C[0,1]$ of continuous functions is complete

Let $C[0,1]$ denote the metric space of all continuous functions $f:[0,1] \rightarrow \mathbb{R}$, with the metric $d(f,g)=\sup |f(x) - g(x)|$ for $0\leq x \leq 1$ Show that $C[0,1]$ is a complete ...
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2answers
71 views

Rigorous proof for a limit of a function

Consider the following expression: $$\lim_{(x,y) \to (0,0)}g(x,y)=\ln(1+3x+4y+x^2+y^4)$$ I have learned in the past that if a function is continuous, I can simply substitute the points into the ...
2
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1answer
55 views

Is this map from $\mathbb{R}$ to $[0,\infty)$ continuous?

Let $f\colon\mathbb{R}\to\mathbb{R}$ be defined as $$ f(x) = \begin{cases} 0 & \text{if $x\leq 0$,} \\ x & \text{if $x > 0$.} \end{cases} $$ Then let $(a,b)$ be an open interval ...
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1answer
51 views

What is an example of a continuous function where the derivative at $x=0$ does not exist? [closed]

What is an example of a continuous function where the derivative at $x=0$ does not exist? This is a word problem, and I don't know how to approach this because it is an unusual question. Any ...
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4answers
66 views

Given a derivative function, a coordinate on original function, can we find a certain limit?

True or false? If we're given: $f'(x) = \frac 1x$ and $f(2) = 9$, then (true or false): $$\lim \limits_{x \to 2}\frac{{\sqrt {f(x)}}-3}{x-2}= \frac 1{12}$$ (Apologize for fractions, couldn't get ...
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1answer
29 views

For which value(s) of a and b is the function defined by, Calculus help?

Could someone aid me in finding the answer to this question? No need to sketch the graph.
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1answer
47 views

Is there exists a $y$ in the interval $(0,1)$ such that $f(y)=f(y+1) ?$

A function $f(x)$ is continuous in the interval $[0,2].$ It is known that $f(0)=f(2)=−1$ and $f(1)=1.$ Which one of the following statements must be true$?$ Options are $:$ There exists a $y$ in ...
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1answer
52 views

A function that has both partial derivatives but is not differentiable or even continuous

Let $f(x,y) = \dfrac{4xy(x+y)}{(x^2+y^2)} $, $f(0,0)=0$. Show that if f(x, y) were differentiable at (0, 0), then $\lim _{(h,h)\rightarrow (0,0)}f(h,h)$ = 0. Finally, show that $\lim ...
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1answer
72 views

Continuous extension of $\int_\mathbb{R} dt\, e^{-t^2}/(t-z)$ from $\operatorname{Im} z < 0$ onto $\mathbb R$

I am asked to show that the continuous extension of $$ F(z) = \int_{-\infty}^{\infty} dt\, \frac{e^{-t^2}}{t-z}, \quad \operatorname{Im} z < 0 $$ onto $\mathbb R$ is given by $$ ...