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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Why is continuity needed to substitute value of derivative inside Riemann-Stieltjes Integral?

Given $f$ increasing on $[a,b]$, $g(x)\in R(\alpha)$ on $[a,b]$, $\alpha \in C([a,b])$ $$ \beta(x)=\int_a^xg(z)d\alpha(z) \text{ on [a,b]} $$ Why is the additional assumption $f$ continuous and what ...
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37 views

Show that $f^{-1}$ is continuous

Let $E$ and $F$ two normed vector spaces, $A \subset E$ compact, $B \subset F$ and $f: A \to B$ is a bijective continuous function. As $f$ is bijective, we can defining the inverse function ...
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2answers
15 views

Show that $Gr(f)$ is compact

Let $A \subset \mathbb{R}^n$ a compact and $f : A \to \mathbb{R}^m$ a continuous function. Let the graph of $f$ $$Gr(f) = \{(x,f(x) : x \in A)\}.$$ Show that $Gr(f)$ is compact. My proof : ...
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1answer
16 views

Definition of continuity up to the boundary

Let $\Omega \subset \mathbb{R}^n$ be open and bounded. What does it mean $f\in C(\bar{\Omega})$, i.e. what does it mean $f$ to be continuous at $x \in \partial \Omega$, maybe $$\forall \epsilon >0 ...
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69 views

Is there a nice open set proof that multiplication is continuous?

For students in a first course in analysis or topology, proving that certain function are continuous can be very tricky. However, some proofs which are difficult for students to prove using the ...
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1answer
38 views

Piece wise function continuity [on hold]

Find all values of $a$ and $b$ so that the following function is continuous for all value of $x$. ($x\in\Bbb R$). $$ f(x)=\begin{cases}-3a+4x^5b&\text{when }x\le -1\\ ax-2b&\text{when ...
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2answers
35 views

Map from circle to real line

I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that: ...
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1answer
18 views

Continuity proving of function with delta-epsilon

Prove continuity of function with the delta-epsilon definition in point $x_o=0$ $$f:\mathbb{R}\rightarrow \mathbb{R}$$ $$f(x) = \begin{cases} x^2+1, & x \in \mathbb{Q} \\[2ex] 2^x, & x \in ...
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1answer
41 views

Complex Continuity [on hold]

Is the function $f$, defined by $$ f(z) = \begin{cases} \frac{z^2+2iz-1}{2z^2+iz+1} & \text{ if } z \not \in \{-i\}\\ 0 & \text{ if } z = -i \end{cases}$$ continuous at $−i$? Explain your ...
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2answers
49 views

How can I show that this function is discontinuous at the point $x=1$?

Suppose you had the function $$ f(x) = \; \text{ the integer part of } x $$ I wish to show that this is not continuous at the point $x=1$, which I will try to do by showing that $\lim_{x \rightarrow ...
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46 views

is $x\sin(\frac{1}{x^2})$ uniformly continuous on $(0,1]$?

Is $x\sin(\frac{1}{x^2})$ uniformly continuous on $(0,1]$? I am really unsure how to start this one off. I have done checks for "similar" functions like: $f=\sin(\frac{1}{x}), x\in(0,1]$, by using ...
2
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1answer
32 views

Show $\cos(x^2)/(1+ x^2)$ is uniformly continuous on $\Bbb R$.

now here's how I did proceed. By definition a function $f: E →\Bbb R$ is uniformly continuous iff for every $ε > 0$, there is a $δ > 0$ such that $|x-a| < δ$ and $x,a$ are elements of $E$ ...
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2answers
54 views

Is $\frac{1}{\sin x}-\frac{1}{x}$ uniformly continuous on $(0,1)$?

So I am tasked with finding whether $\frac{1}{\sin(x)}-\frac{1}{x}$ is uniformly continuous on the open interval $I=(0,1)$. To look at the "simple" ways to prove it first: I obviously can't extend ...
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23 views

Continuous function of bounded variation that is non-monotone? [on hold]

Construct a continuous function of bounded variation on the interval [0,1] which is not monotone in any subinterval. We can follow the pattern of the Cantor-Lebesgue function loosely. For example, at ...
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1answer
40 views

Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range

I was studying the Inverse function theorem when I came across the following problems : (Let the closed set $V$ i.e the range have non-empty interior) Does there exist a continuous onto ...
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10 views

Recovery sequence for semicontinuous functions

I have seen that the next statement holds (if my memory is not wrong) in a certain book. (I forgot which book this is.) For a lower semicontinuous function $f:(0,1)\to\mathbb{R}$ and a given ...
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2answers
27 views

differentiability of a function

Let $f$ be a continuous function on an open interval in $\mathbb R$ such that $|f|$ is differentiable. Can we show that $f$ is differentiable? I can get several examples of non-differentiable $f$ if ...
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17 views

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$?

Why are $F(p) := \sup_{x∈[0,1]}{|p'(x)|}$ and $G(p) := \sup_{x∈[0,1]}{|p(x)|}$ both continuous functions of the polynomial $p$, which is finite and of degree at most $d$ ? Continuity of a function ...
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1answer
11 views

Why is the inf-convolution of lower semicontinuous functions continuous?

I'm confusing now about the continuity of inf-convolution. I understand that the inf-convolution of lower semicontinuous functions is semiconcave and so it's locally Lipschitz continuous (in ...
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0answers
10 views

Why does $y(s)$ continuous imply that $f(s)$ with $f_l (s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum \max\{0,z_l(s)\}}$ is continuous?

Let $z:\triangle^{L-1}\to \mathbb{R}^L$ be continuous. Define $f:\triangle^{L-1} \to \triangle^{L-1}$ be defined component wise as $$ f_l(s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum_{l=1}^L ...
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16 views

Linearizing Euler's continuity equations

I have three Euler equations (for a polytropic gas) and I'd like to linearize them to get 2 other equations. Any help would be appreciated! The 3 initial equations are: 1) ...
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2answers
78 views

Suppose that $f(0)=f(2\pi)$. Show that there exists an x such that $f(x)=f(x+\pi)$.

I am supposed to show that there exists an $x$ in the interval $[0,\pi]$ such that $f(x)=f(x+\pi)$ by considering another function $g:[0,\pi] \to \mathbb{R}$ defined by $g(x)=f(x)-f(x+\pi)$. Should I ...
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2answers
19 views

If $Z(f)$ is the zero set, prove that $Z(f)$ is closed

Introduction: Exercise from Principles of Mathematical Analysis, third edition (Rudin), page 98. Exercise: Let $f$ be a continous real function on a metric space $X$. Let $Z(f)$ (the zero set of ...
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36 views

Derivation with Euler's Equations

I have three equations as follows (for a polytropic gas): 1) $\displaystyle\quad\frac{\partial\rho}{\partial t}+\nabla\cdot\rho \mathbf{u} = 0$ 2) $\displaystyle\quad\rho \left( \frac{\partial ...
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4answers
57 views

Show that a continuous function f either has a root hence $f(c)=0$ or $|f(x)|> e$ for $e>0$.

From what I understand, I am being asked to show that a function $f$ on an interval $[a,b]$ either has a root $c$ such that $f(c)=0$ or it does not have a root hence $|f(x)|\ge e$. Should I apply the ...
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2answers
40 views

Continuity of a function with complex variables

How could I show if or not the following piece-wise defined function is continuous at the point $z=-i$? $$f(z)=\left\{ \begin{matrix} \frac{z^2+2iz-1}{2z^2+iz+1}, & z \neq -i \\ 0, & z=-i ...
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Prove that a function is continuous (square integrability)

I need help for the following proof of continuity: Let $E=L_2([t_0,t_1],\mathbb R)$ be a Hilbert space of square-integrable real-valued functions on $[t_0,t_1]$. Let ...
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2answers
54 views

proof that an arbitrary homeomorphism $h: B_{1}[0] \rightarrow B_{1}[0]$ maps $S^n$ to $S^n$

Intuitively this proposition seems true, but I've been told that is not a trivial thing to prove. Is there any simple proof (or counter-example) for the proposition: Consider the closed ball of ...
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2answers
50 views

Proving continuity with epsilon delta

I have the function $f:\mathbb{R}\rightarrow \mathbb{R}\:\:f\left(x\right)=x^2-3x$ and it asks me to prove continuity in point $\:x_o=0$ using the epsilon-delta definition. I know that in order to do ...
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1answer
36 views

EDITTED: Find all values of $a$ and $b$ so that $ax^n+b\cos\left(\frac{x}{n}\right)$ is Cauchy.

For each $n\in\mathbb{N}$ let $$f_n(x)=ax^n+b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1].$$ Find all values of $a$ and $b$ for which $(f_n)$ is a Cauchy sequence in $C[0,1]$, the space of ...
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1answer
35 views

Problem involving polynomial and arbitrary continuous function

Let $f\in C^4[0,1]$ and $p$ a polynomial of degree $3$. Suppose: $$f(0)=p(0),\quad f'(0)=p'(0),\quad f(1)=p(1),\quad f'(1)=p'(1)$$ Show that for each $x\in [0,1]$ there exists $\xi\in [0,1]$: ...
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1answer
25 views

Continuous mapping theorem to show $g(x_n)$ to $g(x)$ not converges.

Let $X_n$ be a random variable sequence, such that $P(X_n=1)=1/n$ and $P(X_n=1/n)=1-(1/n)$. Let g be a function, such that $ g(x)= 0$ if $ x\le0$, and 1 if $x>0$ Show that $g(X_n)$ not converges ...
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1answer
19 views

wrong proof of “locally lipschitz implies continuity”

I think that I've proved that locally lipschitz implies continuity on metric space. But something must be wrong: Let $(\mathfrak{X},d_1)$ and $(\mathfrak{Y},d_2)$ be metric spaces. If $\varphi ...
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37 views

Wrong result: a continuous function has zero $p$-variation, for every $p$. Where's the error?

Let $\Pi_n$ be a sequence of partitions with $|\Pi_n| \to 0$. Then the $p$-variation of a continuous function $g$ along the partitions $\Pi_n$ is defined as $$V_T^p(g) = \lim_{n \to \infty} V_T^p(g, ...
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1answer
16 views

Prove continuity of function

So I have the function: $$f:\left[-1,2\right]\:\bigcup \left\{3\right\}\rightarrow \mathbb{R}$$ $$f\left(x\right)\:=\:x,\:for\:x\in \left[-1,2\right]\:and\:f\left(x\right)=7,\:for\:x\:=\:3$$ And I ...
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12 views

Showing continuity of a real valued function [duplicate]

Let $S=[0,1)\cup [2,3]$ and let $f:S\to \mathbb{R}$ be strictly increasing such that $f(S)$ is an connected subset of $\mathbb R$. How to show that $f$ is continuous? $f(S)$ is connected means it is ...
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1answer
38 views

Continuous function $h \colon [0, 1) → \Bbb R$ which is bounded but does not attain either of its bounds.

I am trying to find a continuous function $h \colon [0, 1) →\Bbb R$ which is bounded but does not attain either of its bounds. I'm having no luck so any tips would be great thanks.
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29 views

Why can we calculate the supremum of operator norm over unit circle?

I know that to check whether a linear operator is continuous or not we have to check if the operator norm is bounded. $$T: V\to W$$, $$\vert\vert \ T \vert\vert= \sup_{f \in V}\frac{\vert\vert \ Tf ...
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1answer
36 views

Average of Monte Carlo simulations of continuous functions again continuous?

I hope the following question is clear: Suppose, we have a continuous functions $f:\mathbb{N}^2 \rightarrow \mathbb{N}$. Now, suppose we run Monte Carlo simulations on the function, where the input ...
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1answer
31 views

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 these statements must be true?

(A) There exists a $y$ in the interval $(0,1)$ such that $f(y)=f(y+1)$. (B) For every $y$ in the interval $(0,1), f(y) = f(2−y)$. (C) The maximum value of the function in the interval $(0,2)$ is ...
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1answer
28 views

Increasing real valued function whose image set is connected

Let $S = [0,1) \cup [2,3]$ and $f\colon S \rightarrow \mathbb R$ be such that $f(S)$ is connected . Which of the following are true: a) $f$ is discontinuous exactly at one point. b) $f$ is ...
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2answers
31 views

Prove that $f(z)=f(re^{i \theta})= \sqrt{r}e^{i \frac{\theta}{2}}$ is discontinuous

I would like to prove thatthe multiform function $f(z)=f(re^{i \theta})= \sqrt{r}e^{i \frac{\theta}{2}}$ is not continuous for $z \in (- \infty, 0)$ and $\theta \in (-\pi, \pi]$. I think I could use ...
3
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1answer
29 views

Show that $d(E,F) > 0$ - Is $E \times F$ is a compact set here?

Let $d(E,F)=\inf\{|z-w| : z \in E, w \in F\}$ where $E \subset \mathbb{C}$ is compact and $F \subset \mathbb{C}$ is closed such that $E \cap F = \emptyset$. Show that $d(E,F) > 0$; namely, ...
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13 views

Given $f:[a,b] \rightarrow R$, if $x'$ is a local minimum of $f$ and $x'<b$ then there exists a sequence $x_n$ converging to $x'$ with $x'<x_n<b$

I'm trying to understand the demostration of the folowing lemma: Is a function $f:[a,b] \rightarrow R$ differentiable in a local extrema $x'$ then $f'(x')=0$ Demostration: $x'$ is a local ...
2
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1answer
22 views

Show with the definition that the function $f:\mathbb{R^3} \to \mathbb{R^2}$ defined as $f(x,y,z) = (x^2-y^2+z, 4|xy|)$ is continuous

Show with the definition that the function $f:\mathbb{R^3} \to \mathbb{R^2}$ defined as $f(x,y,z) = (x^2-y^2+z, 4|xy|)$ is continuous. Let $\alpha=(x,y,z), \alpha_0=(x_0,y_0,z_0)\in ...
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0answers
25 views

Show with the definition that the function $f:\mathbb{R^3} \to \mathbb{R^2}$ defined as $f(x,y,z) = (x^2-y^2+z, 4|xy|)$ is continuous

Show with the definition that the function $f:\mathbb{R^3} \to \mathbb{R^2}$ defined as $f(x,y,z) = (x^2-y^2+z, 4|xy|)$ is continuous. Let $\alpha=(x,y,z), \alpha_0=(x_0,y_0,z_0)\in ...
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1answer
68 views
+50

$f\in C^1(\mathbb R)$ , having finitely many zeroes and $f'$ changes sign at exactly two of these points , solutions of $f(x)=y$ for given $y$?

Let $f:(0,1) \to \mathbb R$ be a continuously differentiable function having finitely many zeroes and $f'$ changes sign at exactly two of these points , then is it true that for any $y \in \mathbb R$ ...
2
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1answer
45 views

Spivak's Calculus exercise related to $\sup$ and $\inf$

This is the exercise 4.b of chapter 8 from Spivak: Suppose that $f$ is continuous on $[a,b]$ and that $f(a)<f(b)$. Prove that there are numbers $c$ and $d$ with $a \leq c <d \leq b$ such that ...
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1answer
39 views

If $f:[a,b]\to\mathbb{R}$ is continous at $s$, then if $s<b$ can I find an $x \in [a,b]$ with $s<x<s+\delta$? [closed]

If $f:[a,b]\to\mathbb{R}$ is continous at $s$ where $s \in [a,b)$, then if $s<b$ can I find an $x \in [a,b]$ with $s<x<s+\delta$? $\delta$ is $\delta$ of the epsilon-delta definition of ...
0
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1answer
27 views

Showing a set is equicontinuous and uniformly bounded

Let $C[0,1]=\left\lbrace \text{real-valued, continuous functions on [0,1]} \right\rbrace$, and suppose $F$ is precompact in $C([0,1])$. I need to prove that $F$ is uniformly bounded and ...