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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$f(x)=\sum_{n=0}^{+ \infty} \frac{(-1)^n}{(n!)^2}\left( \frac{x}{2}\right)^n $ is continuous

Let \begin{align} f: \begin{cases} \mathbb{R} &\longrightarrow \mathbb{R} \\ x & \longmapsto \displaystyle \sum_{n=0}^{+ \infty} \frac{(-1)^n}{(n!)^2}\left( \frac{x}{2}\right)^n\end{cases} ...
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133 views

To what extent is a function that is analytic on the unit disk determined by its boundary values?

Suppose we have a function that is analytic on the open unit disk. Suppose we have a continuous function on the boundary of the disk that maps each point on the boundary of the disk to its conjugate. ...
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72 views

Determine all continuous functions

I don't really understand what is meant by the following problem when it is said "determine all continuous functions:" Let $X$ be a metric space, and $Y$ be a discrete metric space. Determine all ...
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1answer
49 views

Continuity of functions from complex numbers

i have a question about continuity. Suppose i have a function from $\Bbb{C}$ into a Banachalgebra $A$ for example $r\mapsto exp(ra)$ for a fixed $a\in A$. Do we have to prove continuity by ...
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68 views

Continuous functions on intervals.

Problem Statement:$$ \text{Let}\ I = [a,b]\ \text{and let}\ f: I \rightarrow \mathbf{R}\ \text{and}\ g: I \rightarrow \mathbf{R}\ \text{be continuous on}\ I.\\ \text{Show that the set}\ E := \{x \in ...
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1answer
58 views

Continuity and openness proof

I need some help proving the following theorem. My professor said that it was a "local" version of an important theorem: Suppose f:X→Y and a ∈X. The function f is continuous at a if and only if for ...
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2answers
112 views

“Rising sun” function

Let $f:[0,1] \to \mathbb R$ be bounded and $$ f_\odot :[0,1] \to \mathbb R: x \mapsto \sup \{f(y) : y \in [x,1] \} $$ This is well defined since $f$ is bouned. Claim: If $f$ is continuous then ...
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1answer
73 views

Prove that d(.;A) is continuous

I need help with the following proof, which my professor added for practice (but not as homework). I am completely lost here. Let $A$ be a nonempty subset of a metric space $X$. Define $d(\cdot ,A) : ...
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1answer
328 views

Dini's continuity vs Holder continuity

(listed items are just the definitions, you can skip to "Clearly" if you are familiar with them) Let $E \subset \mathbb{R}^N$ and let $f \colon E \to \mathbb{R}.$ The modulus of continuity of $f$ is ...
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27 views

Where does the value 1200 come from in this equation for a uniformly continuous function

This example came from the $\epsilon-\delta$ criterion section of my book: $f(x)=x^{3}$ for $x$ in $[0,20].$ Then the function $f$ is uniformly continuous. To see this, observe that for all $u$ and ...
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1answer
170 views

Let $f:[0,1]→\mathbb{R} $with $f′(x) $continuous. It is known that $\int_{0}^{1} f(x)dx=0$.

Let $f:[0,1]→\mathbb{R}$ with $f'(x)$ continuous. It is known that $∫_0^1 f(x) dx=0$. Prove that $∀α∈[0,1]$, $$|\int_{0}^{\alpha} f(x) dx |≤ \frac{1}{8} sup_{(0≤x≤1)}|f'(x) |$$ My answer so far ...
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1answer
390 views

Finding points of continuity on piecewise function

For what values of $a$ and $b$ is the function continuous at every $x$? $$\displaystyle f(x)=\begin{cases} -1 & \text{if }\;\; x \leq -1\\ ax+b & \text{if }\;\; -1<x<3\\ 13 ...
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4answers
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What is a continuous extension?

The continuous extension of $f(x)$ at $x=c$ makes the function continuous at that point. Can you elaborate some more? I wasn't able to find very much on "continuous extension" throughout the ...
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78 views

Real Analysis Derivative

Consider the function $f\colon\mathbb R\to\mathbb R$ where $f$ is defined by $$f(x)= \begin{cases} x^b\sin(1/x), &\text{if $x>0$};\\ 0,&\text{if $x<=0$.} \end{cases}$$ Prove that the ...
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120 views

To find a Coordinate Patch About a Point in Euclidean Subspace.

I have been trying to settle this question for a long time now and it is very important for me to solve this. Let $p, q\in \mathbb R^2$ be points such that $p$ and $q$ are linearly independent (when ...
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1answer
344 views

Prove $h(x)=\sqrt{x^2-1}$ continuous by $\epsilon,\delta$

Proof: Let $h\colon (1, \infty)\to \Bbb R$ be a function. Let $h(x)= \sqrt{x^2-1}$. Let $\epsilon>0$ be arbitrary. Let $x_0\geq 1$ be arbitrary. Suppose $x_0 > 1$. Let ...
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1answer
112 views

Continuity of a piecewise function on the irrationals.

I have a function defined as follows: Let $t=\frac{p}{q}$ be fully reduced for $t\in \mathbb Q$ $$ f(t)=\left\{ \begin{array}{lll} 1/q & \text{if} & x\in\mathbb Q\\ 0 & \text{if} & ...
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3answers
149 views

Can there exist a function with discontinuity at Cantor's Set union Z?

I know there can't exist function with discontinuities only at irrational points,since cantor set is also uncountable like irrational numbers,I thought that the answer is no. Also if yes can you give ...
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45 views

Showing that if $x_k \rightarrow x \implies f(x_k) \rightarrow f(x)$, then $f^{-1}(C)$ is closed for any closed set

As part of proofs on continuity, I should show that (i) $\forall x \in \mathbb{R}^n,$ if $x_k \rightarrow x \implies f(x_k) \rightarrow f(x)$ implies (ii) $f^{-1}(C)$ is closed for any closed set ...
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1answer
80 views

conclusion about function in topological space

Consider the following topological space: $\tau= \{U\subseteq R: 1\notin U\} \cup \{R\}$ and the following function: $f: (R, \tau)\to (R, \tau)$ I have already proved that: 1) if $f(1)=1$, then ...
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91 views

Uniformly continuous bijection from $X$ to the Cantor set

Let $X$ be a metric space, and $C$ be the Cantor set (equipped with the standard topology). Let $f: X\to C$ be a uniformly continuous function. Assume that $f$ is a bijection. Does it follow ...
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1answer
105 views

If $|f|$ is Hölder continuous, what about $f$?

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function such that $|f|$ is Hölder continuous with exponent $0<\alpha\leq1$. Does it follow that $f$ is also Hölder continuous with ...
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2answers
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Proving the continuity of a difference of functions.

Prove that if $f$ and $g$ are continuous at $x=a$, then $(f-g)$ is continuous at $x=a.$ I have $|f(x)-f(a)-g(x)+g(a)| = |(f(x)-g(x))-(f(a)-g(a))|$ so far. I wanted to use the triangle inequality on ...
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47 views

Proof this function is constant

I have the following topological space: $\tau= \{U\subseteq R: 1\notin U\} \cup \{R\}$ and the following application: $f: (R, \tau)\to (R, \tau)$ I have already proved that if $f(1)=1$, then $f$ ...
0
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1answer
24 views

Continuity- Image of a function

I have the following topological space: $\tau=$ {$U\subseteq R: 1\notin U$} U {$R$} and the following application: $f: (R, \tau)\to (R, \tau)$ I have to see that if f(1)=1, then f is continuous. ...
2
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2answers
69 views

Continuity only at $x_0=0$

Consider the function \begin{align}f: \mathbb{R} &\longrightarrow \mathbb{R} \\ x: & \longmapsto \begin{cases} x, \ x \in \mathbb{Q} \\ 0, \ x \notin \mathbb{Q} \end{cases} \end{align} ...
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1answer
70 views

Continuity of Green's function

Suppose $\Omega \subset \mathbb C$ is a region (open and connected set) and let $$g(z,z_0)=G(z,z_0)-\log|z-z_0| $$ be its Green's function with pole at $z_0 \in \Omega$. Here $G(z,z_0)$ is the ...
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633 views

Finding value(s) of a for which f is continuous

A function $f$ is defined as follows: $$f(x)=\begin{cases}\sin x&\text{if }\;x\leq c,\\ax+b&\text{if }\;x>c,\end{cases}$$ where $a,b,c$ are constants. If $b$ and $c$ are given, find all ...
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1answer
79 views

Prove a function is not uniformly continuous. [closed]

Use the definition of uniform continuity to prove the function G(x) = x^3 is not uniformly on [0, infinity).
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1answer
49 views

Proof for the property of a fixed point of $f$.

Suppose that $f \colon [a, b] \to [a, b]$ is continuous. (Note that the range of $f$ is a subset of $[a, b]$) Prove that there exists at least one point $x \in [a, b]$ such that $f(x) = x$. A point ...
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1answer
387 views

Prove discontinuity of f(x)=-x+5 using epsilon delta

Let $f(x)=\begin{cases} 2x + 3&\text{ for }x\geq 1,\\ -x+5 &\text{ for }x<1. \end{cases}$ $f$ is continuous from the right at $x\geq1$. The proof would be: ...
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3answers
64 views

Advantages to continuity at a point

A scalar field $f : \mathbb{R}^n \to \mathbb{R}$ is said to be continuous at a point $\boldsymbol{a}$ if $$ \lim_{\boldsymbol{x} \to \boldsymbol{a}} f(\boldsymbol{x}) = f(\boldsymbol{a}) $$ So in ...
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1answer
71 views

Is the derivative in $C(\overline{\Omega})$?

Let $\Omega\subset\mathbb{R}^2$ a bounded domain and consider $u\in C^2(\Omega)\cap C(\overline{\Omega})$. I would like to know if then $$ 1+\frac{\partial u}{\partial x}\in ...
0
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1answer
77 views

B-spline parameterization and derivatives

I have a question regarding the re-parameterisation of a B-spline. Some info: The B-spline is of order 4 (degree 5), hence $C^3$ continuity There is no knot multiplicity The end conditions are not ...
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1answer
58 views

Prove that $\bar{h}:Y\to Z$ is a continuous function.

I've run into this rather tricky question (to me at least). Let $(X,\mathcal{T})$ be a topological space and let $Y$ be another set and let $f:X \to Y$ be a surjective function, and equip $Y$ with ...
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1answer
119 views

$\epsilon$-$\delta$ Verification of a Lipschitz Function

A function $f:D\rightarrow \mathbb{R}$ is said to be a Lipschitz function provided that there is a nonnegative number $C$ such that $|f(u)-f(v)|\le C|u-v|$ for all $u,v\in D$. Show that a Lipschitz ...
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3answers
102 views

Define $f(x)=x^3$ for all $x$. Verify the the $\epsilon - \delta$ criterion at each point $x_0$.

Following Clayton's advice, for $\epsilon>0$, we must find a $\delta>0$ such that $|x^3-x_0^3|<\epsilon$ if $|x-x_0|<\delta$. $|x^3-x_0^3|=|x-x_0||x^2+x_0x+x_0^2|<\epsilon$ I'm still ...
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1answer
52 views

Let $f \colon \Bbb R \to \Bbb R$ be continuous such that $f(a) \ge b,\, f(b) \le a$

Let $f \colon \Bbb R \to \Bbb R$ be continuous such that $f(a) \ge b,\, f(b) \le a$. Then $\exists x \in [a,b]$ such that (A)$f(x)=0$ (B)$f(x)=x$ (C)$f'(x)=0$ ...
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Calculating $\lim_{x \to 0}\frac1x \int_{0}^{x}f(y)dy$ when $f$ is continuous

I was trying the following problem which is : Let $f \colon \Bbb R \to \Bbb R$ be a continuous function such that $$\lim_{x \to 0}f(x)=a .$$ Then $$\lim_{x \to 0}\frac1x \int_{0}^{x}f(y)dy $$ is ...
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1answer
101 views

Proving that $f(x,y) = \frac{xy^2}{x^2 + y^2}$ is a continuous function using epsilon-delta.

THE QUESTION: Use the metric $(x,y)$ = $\rho(x,y)=|x-y|$ for the reals and use the metric $\rho((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ for the plane. Define $f:R\times R \to R$ as ...
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1answer
82 views

In metric spaces, is a function uniformly continuous iff $\delta$ depends on $\varepsilon$?

Most book examples end with an expression for $\delta$ that depends on $\varepsilon$ when proving uniform continuity. What I am wondering is whether a function can be uniformly continuous as long as ...
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71 views

$\epsilon-\delta$ Verification

Define $f(x)=\sqrt{x}$ for all $x\ge 0$. Verify the $\epsilon-\delta$ criterion for continuity at $x=4$. Hint: first show that for $x\ge 0$ and $x_0> 0$, ...
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17 views

Continuity type problem involving a limit?

I am trying to solve this problem involving a limit / continuity. https://www.dropbox.com/s/aglohigapdfalny/Screenshot%202013-10-30%2020.22.40.png I've set the equations equal to each other and ...
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64 views

Prove B is a closed subset of X given the f and g are continous?

Let $(X;\rho)$, $(Y;\sigma)$ be metric spaces. Let $f,g : X \to Y$ be continuous. Prove that the set $B=\{x\in X: f(x)=g(x)\}$ is a closed subset of $X$
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258 views

True or False: Continuous Functions (Extreme Value Theorem)

Do my justifications seem appropriate? a.) Every function $f:[0,1]\rightarrow \mathbb{R}$ has a maximum. True; if $f:[0,1]\rightarrow \mathbb{R}$, $f$ wil be closed and bounded above, so it will ...
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153 views

Prove that the identity map $(C[0,1],d_1) \rightarrow (C[0,1],d_\infty)$ is not continuous

$$d_\infty = \max|x_i - y_i|$$ $$d_1 = \sum_{i=1}^n |x_i - y_i|$$ The first part of this question was to prove that the identity map $$(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$$ is continuous, ...
5
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1answer
73 views

A function $f:I\times I\longrightarrow I$ continuous in each variable, where $I=[0,1]$

Let $f:I\times I\longrightarrow I$ be continuous in each variable, where $I=[0,1]$. Can we show there exists one point where the function is continuous?
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1answer
50 views

Extensions of continuous bounded functions

If $u:U \rightarrow \mathbb{R}^{n}$ is bounded and continuous can $u$ always be extended such that $u \in C(\bar{U})$? and is $u$ uniformly continuous?
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35 views

Introduction to Analysis: Bisections

I feel as though I may be over thinking this problem, at the same time I feel like I may be under thinking it. Use a bisection argument to prove that if $f:[a,b] \rightarrow \mathcal{R}$ is ...
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2answers
38 views

Continutiy and Contradiction: I'm not sure if I have the correct negation.

Suppose that the function $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous at the point $x_0$ and that $f(x_0)>0$. Prove that there is an interval $I\equiv(x_0-\frac{1}{n},x_0+\frac{1}{n})$, ...