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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4
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3answers
145 views

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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0answers
9 views

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 $ ?
4
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3answers
60 views

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 ...
2
votes
1answer
25 views

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.
0
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0answers
32 views

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 ...
0
votes
2answers
46 views

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 ...
0
votes
2answers
29 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 ...
1
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0answers
61 views

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 ...
1
vote
3answers
66 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 ...
0
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0answers
28 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} \}\\ ...
1
vote
1answer
34 views

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 ...
0
votes
2answers
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} ...
2
votes
2answers
37 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 ...
3
votes
2answers
26 views

Multivariable continuity in ball

How do you show continuity for ball-based functions such as $$f:B[(0,0),1)]\rightarrow\mathbb{R}, \space f(x,y) = \sqrt{(1-(x^2+y^2)}$$
3
votes
2answers
34 views

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 ...
0
votes
1answer
37 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 ...
1
vote
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 ...
2
votes
2answers
193 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 ...
2
votes
2answers
87 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
votes
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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votes
1answer
54 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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votes
4answers
69 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 ...
0
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1answer
30 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.
0
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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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votes
1answer
63 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 $f(x, y)$ is differentiable at $(0, 0)$, then $\lim _{(h,h)\rightarrow (0,0)}f(h,h) = 0$. Finally, show that $\lim ...
1
vote
1answer
73 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 $$ ...
0
votes
2answers
46 views

How to prove that $x\mapsto x^{\frac35}$ is continuous [closed]

Trying to teach myself calc and I know what continuity is but how exactly do we write a formal proof proving that a function is continuous. Say the function $f(x) = x^{\frac35}$. How would I write ...
0
votes
1answer
44 views

How to quickly check the continuity of a given function?

I would like to know if there was a given function, $f(x,y) = y\sqrt x$,how would I quickly check if the function is defined and continuous for $y(0) = 0$? Note: Just wanna check the condition for ...
2
votes
1answer
153 views

Is a path homotopy equivalence class a path component?

Let $X$ be a topological space and let $\Omega=\Omega (X;a,b)$ be a path space of $X$ from $a\in X$ to $b\in X$ with a compact-open topology. For any $f\in \Omega$, we can consider a homotopy ...
2
votes
1answer
26 views

Condition on subsets of normed linear space such that “every real valued continuous function on the subset is uniformly continuous” imply boundedness

If $A$ is a connected subset of a real normed linear space such that every real valued continuous function on $A$ is uniformly continuous , then is it true that $A$ is bounded ? If not , then what if ...
0
votes
1answer
28 views

Solving the boundary value problem?

Let $G(x,y)$ be the Green's function of the boundary value problem $$[(1+x)u']'+(\sin{x})u=0,~x\in[0,1],~u(0)=u(1)=0.$$ Then, the function $g$ defined by $$g(x)=G(x,\dfrac{1}{2}),~x\in[0,1].$$ is ...
2
votes
1answer
39 views

Give an example in which $a_0=a_1<a_2=a_3<a_4=a_5<a_6=\cdots$ for the bisection method. [closed]

Give an example in which $a_0=a_1<a_2=a_3<a_4=a_5<a_6=\cdots$ for the bisection method. I don't see how to make an example of this form. I'm sure there is something simple/elegant that does ...
0
votes
0answers
70 views

Alternative proof for dominated convergence theorem without using Fatou's lemma?

The conclusion of dominated convergence theorem is that $||f-f_n||_{L^1}\to 0$ as $n\to\infty$. After showing that $f_n\to f\in L^1$, why is it not possible to use the continuity of norm in order to ...
2
votes
1answer
52 views

Continuity of the Fourier transform of a measure

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ then $$x \mapsto \hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle } \Bbb d \mu _{(y)}$$ is ...
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0answers
17 views

Laplace Transform of uniformly convergent series

Let $\sum_{n=1}^{\infty} f_n(x)$ be a uniformly convergent series of functions each of which has a laplace transform defined for $s \geq \alpha$. Show that $f(x)=\sum_{n=1}^{\infty} f_n(x)$ has a ...
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votes
1answer
29 views

Existence of minima between two maxima

Let $f:[a,b]\subset \mathbb{R}\rightarrow \mathbb{R}$ be a continuous function. Let $x_1,x_2 \in (a,b),x_1<x_2$ be local maxima of the function f, i.e., $\exists \delta_1>0\ such\ that\ \forall ...
1
vote
3answers
40 views

Can the function $x\mapsto \frac1{x+1}$,be defined over $\mathbb R$?

I am getting confused between what it means for a function to be defined over a domain and continuity. My understanding is that a function being defined over an interval or domain means that all the ...
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3answers
40 views

How to draw the graph of a function of $x$.

$$y=\begin{cases} x^3 & x\le 1 \\ x & x \ge 1\end{cases}$$ Both are continuous at $x=1$. But not differentiate at the point. Is the graph right?
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1answer
41 views

Show that $|c_n-c_{n+1}|=2^{-n-2}(b_0-a_0)$. [closed]

Let $c_n={1\over 2}(a_n+b_n)$, $r=\lim_{n\to \infty}c_n$, and $e_n=r-c_n$. Here $[a_n,b_n]$, with $n\geq 0$, denotes the successive intervals that arise in the bisection method when it is applied to a ...
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2answers
89 views

Continuous Random Variable question, Probability and Statistics

a little help please A couple decide they really want a daughter. So, they decide to start having children and continue until they have their first daughter. Assuming having either a boy or girl is ...
2
votes
1answer
65 views

Continuity and open subsets

I am totally lost with the following proof. I want to show that if a function $f : X \to R$ is continuous with $f(x) \gt 0$ for a particular $x \in X$, where $X$ is a metric space, then we have an ...
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0answers
48 views

If $X$ is a compact Hausdorff space, then $\mathcal{C}(X)$ is reflexive iff $X$ is finite.

Let $X$ be a compact Hausdorff space. Then show that $\mathcal{C}(X)$ is reflexive iff $X$ is finite, where $\mathcal{C}(X)$ is the set of all continuous from $X$ to the base field ($\mathbb{R}$ or ...
4
votes
1answer
52 views

Representation of $\mathbb{R}$, drop continuity assumption, Axiom of Choice.

A representation, for instance, of $\mathbb{R}$ is a group homomorphism $f: \mathbb{R} \to \text{GL}_n(\mathbb{R})$. If we assume that $f$ is continuous, then there is a very nice formula for all such ...
2
votes
1answer
97 views

Derivability and Differentiability

Consider the function $f(x,y) = (x^2y)^\frac{1}{3}$ continuous is $\mathbb{R}^2$ and its partial derivatives: $$\begin{cases} f_x(x,y) = \frac{2}{3}\left(\frac{y}{x}\right)^\frac{1}{3} \\ f_y(x,y) = ...
1
vote
1answer
142 views

nonempty subset $E$ of $R$ closed and bounded iff every continuous real-valued function on $E$ takes a maximum value.

I need to show that a nonempty subset $E$ of $R$ is closed and bounded iff every continuous real-valued function of $E$ takes a maximum value. I believe that "if $E$ is closed and bounded, then ...
4
votes
1answer
43 views

$f :\mathbb{R}\to \mathbb{R}$ with closed fibers sending connected to connected

Let $f:\mathbb{R}\to \mathbb{R}$ be a map with closed fibers that sends connected sets to connected sets. Is it true that $f$ is continuous?
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2answers
80 views

Isn't the Continuity Concept Redundant (up to definitions)? [closed]

The idea of continuity in (real) analysis—and indeed everywhere else it is used: topology, etc.—seems to me superfluous. For if, as I learnt from Stewart (Concepts of Modern Mathematics), it is based ...
4
votes
2answers
222 views

Is a continuous open surjection $f:\mathbb R \to \mathbb R$ a homeomorphism?

I have this problem: If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous, open surjection, must it be a homeomorphism? What about if $f$ was defined from $S \rightarrow S$ instead, where S is ...
1
vote
1answer
48 views

Why is there a problem with the derivative of a removable discontinuity?

I have been told that you cannot derive a function at a removable discontinuity. The derivative is defined as: $$ \lim_{h\to0}\frac{f(x+h) - f(x)}{h} $$ And if there is a removable discontinuity at ...
1
vote
1answer
52 views

How to prove that the inverse of the given continuous function is not continuous

I have the following problem: Consider the identity map $id: C_{max} \rightarrow C_{int}$ where $C_{max}$ is the metric space $C([a,b],\mathbb{R})$ of continuous real valued function defined on ...