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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3
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How do I solve $\lim_{x\rightarrow1}\frac{x+3x^2+5x^3+\cdots+(2n-1)x^n-n^2}{(x-1)}$ in a simple way?

I have to find whether the function $\mathrm f(x)$ defined as $$f(x)=\begin{cases}\cfrac{x+3x^2+5x^3+\cdots+(2n-1)x^n-n^2}{(x-1)}, \;\;&\text{for $x\ne1$}\\[2ex]\cfrac{n(n^2-1)}{3} &\text{for ...
-1
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1answer
18 views

Prove the following lemma

If $f$ is differentiable at $X_o$, then $f(X)-f(X_o)= (d_{x_o} f)(X-X_o)+ E(X)|X-X_o|$, where E is defined in the neighborhood of $X_o$ and $\lim_{X\to X_o}$ $E(X)=E(X_o)=0$ I don't know how to ...
0
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1answer
13 views

Show that this piece-wise function is continuous at 0

Show that this function is continuous at $x=0$: $$h(x) = \begin{cases} \ \ x^2 \quad \text{if} \ x\not\in\Bbb{Q} \\ -x^2 \quad \text{if} \ x\in\Bbb{Q}\end{cases}$$ The answer is pretty clear to ...
0
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14 views

Continuity of the divergence of a static electric field

Let $\rho:\Bbb R^3\to\Bbb R$ be a continuous charge density function. Define the electric field $\vec E:\Bbb R^3\to\Bbb R^3$ by $$\vec E(\vec r)=k\cdot\int_{\Bbb R^3}\rho(\vec{r}')\cdot\frac{\vec ...
1
vote
1answer
43 views

Is $|\sin x|$ a Lipschitz function?

I would like to know how can I prove that the function: \begin{equation} f(x)=|\sin x|, \quad x\in \mathbb{R} \end{equation} is (or isn't) Lipschitz continuous. I studied an example of the funtion ...
0
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1answer
31 views

Show that f is not continuous at a?

Let $a\in \mathbb R$ and $f:\mathbb R\to \mathbb R$ be defined by $f(x) = 1$ when $x > a$, and otherwise $f(x)=0$. Show that $f$ is not continuous at $a$. This problem is in a section on open ...
0
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0answers
17 views

A Question on Continuous Functions over Topological Spaces

Let $f:X \rightarrow X$ be a continuous function on a topological space $X$. Under what conditions is it the case for every subset $A \subseteq X$ that $$A \cap \bigcap_{i=1}^{\infty} f^{-i}((X-A)\cup ...
1
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0answers
28 views

Examine whether the identity map is continuous or NOT.

Consider the space $C[0,1]$. Consider the following metrics $$d_1(f,g)=\int_0^1|f(t)-g(t)|\,dt.$$ $$d_2(f,g)=\left(\int_0^1|f(t)-g(t)|^2\,dt\right)^{1/2}.$$ $$d_3(f,g)=\sup_{0\le t\le ...
0
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0answers
13 views

Hemicontinuity of correspondence

I have a correspondece $f$ of $\mathbb{R}$ in $\mathbb{R}$ that gives me $\textrm{sin}(1/x)$ if $x \neq 0$ and $[-1,1]$ if $x=0$. I have to test whether this function is continuous or not, using ...
1
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0answers
17 views

Computability, Continuity and Constructivism

Triggered by an IMO extremely interesting question & Mathematics Stack Exchange, asked by Dal: Computability and continuous real functions And a link in one of the comments that could have ...
0
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0answers
26 views

Uniform continuity of a continuous function

Let $(X,d);(Y, \rho)$ be metric spaces , $(X,d)$ have nearest point property and $f:X \to Y$ is a continuous function ; then is it true that $f$ is uniformly continuous on any bounded set $A$ in $X$ ...
2
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2answers
33 views

Metric Spaces: Continuous, Unbounded Functions

The following question is from Fred H. Croom's book "Principles of Topology" Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous, unbounded function. Show that there is a number $t_0$ for ...
2
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0answers
26 views

Proving continuity using $\epsilon$ and $\delta$

Here is my first attempt to prove continuity using epsilon and delta: Prove that $f(x)=\sqrt{x}$ is continuous at $p=4$. Unfortunately my book has just some answers and that isn't one of them. I would ...
0
votes
1answer
18 views

Show there is a point c in (a,b) such that f''(x)<0 for x in [a,c) and f''(x)>0 for x in (c,b]

f is twice continuously differentiable on [a,b] and three times differentiable on (a,b), with f(a) = f(b) = 0, f'(a+) = f'(b-) > 0, f'''(x) > 0 for x in (a,b) (NB: a+ means approach a from the ...
3
votes
2answers
41 views

proof : f continuous at a then |f| is continuous at a

Here's my proof, which I am not sure is correct : Assume f is continuous at a $=> \lim \limits_{x \to a} f(x) = f(a)$ $=> \lim \limits_{x \to a} f(x)$ exists $=> \lim \limits_{x \to a} ...
0
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2answers
34 views

Show that $f\left(\lambda \right)=\lambda $ .

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous function and sequence $x_n$ defined as: $x_1=a,$ $a\in \mathbb{R}$ $x_{n+1}\:=\:f\left(x_n\right)$. Also, $\lim _{x\to \infty ...
0
votes
1answer
57 views

Let $f :\mathbb R\rightarrow\mathbb R$ be a function such that $f(x + 1) = f(x)$ for all x ∈ R. Which of the following statement(s) is/are true?

The given options are: (A) f is bounded. (B) f is bounded if it is continuous. (C) f is differentiable if it is continuous. (D) f is uniformly continuous if it is continuous. Any hints on how to ...
2
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0answers
14 views

Uniformly continuous maps between topological groups

Let $G$ be a topological group. For every neighbourhood $U$ of the identity, let $L_U$ be the set of all pairs $(x,y) \in G \times G$ such that $x^{-1} y \in U$. For topological groups $G$ and $H$, a ...
0
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0answers
18 views

Riemann integral and continuity

I have a riemann integrable function on compact set [closed interval in reals]. I want to apply stone-weiestrass theorem from Rudin's. I need continuity of riemann integrable function and I don't know ...
0
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2answers
18 views

Prove existence of a point in continuous function

Let $f(x)$ be a continuous function in $\left [ 0,1 \right ]$ so that for every $x\in \left [ 0,1 \right ]$ there is $f(x)\in \left [ 0,1 \right ]$. Prove that there such a point $x_0\in \left [ 0,1 ...
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2answers
37 views

If $f $ is a continuous function then $\exists a$ such that..

Suppose $ f $ is a continuous function on the interval $[x,y]$; prove $\exists $ $a\in [x,y]$ such that $ f(a)=1/(y-x) \int^y_x fdx $. I can't come up with a neat proof. Theorem: Let $ f $ be a ...
6
votes
4answers
86 views

How to prove this limit is $0$?

Let $f:[0,\infty )\rightarrow \mathbb{R}$ be a continuous function such that: $\forall x\ge 0\:,\:f\left(x\right)\ne 0$. $\lim _{x\to \infty }f\left(x\right)=L\:\in \mathbb{R}$. $\forall ...
0
votes
1answer
23 views

Let $f$ be a continuous function on $[a,b]$ with $f(a) \lt 0 \lt f(b)$. Then there is a largest $x$ in $[a,b]$ with $f(x)=0$.

The following is from Spivak's Calculus. Let $f$ be a continuous function on $[a,b]$ with $f(a) \lt 0 \lt f(b)$. Then there is a smallest $x$ in $[a,b]$ with $f(x)=0$. We show that there is a ...
3
votes
3answers
59 views

Example 4.21 in Baby Rudin: How is the map $f^{-1}$ not continuous at the point $(1,0) = f(0)$?

Let $f \colon [0,2\pi ) \to \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1\}$ be defined as $$f(t) = ( \cos t , \sin t) \ \ \mbox{ for all } \ t \in [0, 2\pi).$$ Then the map $f$ is bijective and ...
0
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1answer
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Theorem 4.20(c) in Baby Rudin: Is every continuous function whose domain is an unbounded subset of $\mathbb{R}$ uniformly continuous?

Here is Theorem 4.20 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Let $E$ be a non-compact set in $\mathbb{R}^1$. Then (a) there exists a continuous function on ...
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1answer
52 views

Theorem 4.20 in Baby Rudin: How is this map not uniformly continuous?

Let $E$ be a bounded, non-compact subset of $\mathbb{R}$, let $x_0$ be a limit point of $E$ such that $x_0 \not\in E$, and let $f \colon E \to \mathbb{R}$ be defined by $$f(x) \colon= \frac{1}{x-x_0} ...
0
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1answer
32 views

Max of $f^2$ in terms of max of $f$?

Is it true that $\max(|f(x)|^2) = (\max |f(x)|)^2$? Where $f: [a,b] \to \mathbb{C}$ and $f$ is continuous.
3
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3answers
65 views

Find all continuous functions $f(x)^2=x^2$

Find all functions $f$ which are continuous on $\mathbb R$ and which satisfy the equation $f(x)^2=x^2$ for all $x \in \mathbb R$. Clearly $f(x)=x, -x, |x|, -|x|$ all satisfy the condition. However, ...
2
votes
1answer
64 views

When does $\lim_{z\to a}f(z)$ exist when $\lim_{z\to a}|f(z)|=L'$?

The following is an excerpt from Silverman's Complex variables with applications discussing the question in the title. However, I don't understand the bolded parts. If $\lim_{z \to a}f(z)=L$, then ...
0
votes
1answer
36 views

f is continuous at a if and only if osc(f,a)=0

So I'm trying to prove this: Let f:A-->R be bounded. Then f is continuous at a if and only if osc(f,a)=0. I have the proof assuming f is continuous and showing osc(f,a)=0. However, I can't prove the ...
2
votes
0answers
21 views

Borel functions and continuous functions [duplicate]

Suppose we have a set $A\subset\mathbb {R}$ and let $f\in\mathcal{B}(A)$ and $g\in\mathcal{B}_b(A)$ (Borel function on $A$ and bounded Borel function on $A$, resp.) Is it possible to approximate $f$ ...
1
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0answers
40 views

Continuity of improper integral $F(y) = \int_1^{+\infty} \sin \frac{y}{x^2} \cdot \sqrt{\ln x} \, dx, \hspace{3mm} y \in \mathbb R$ [on hold]

I have to find out if the following function is continuous $$F(y) = \int_1^{+\infty} \sin \frac{y}{x^2} \cdot \sqrt{\ln x} \, dx, \hspace{3mm} y \in \mathbb R$$ Thanks in advance.
2
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2answers
65 views

Let $f$ continuos in $a\in I$. If $\lim \frac{f(y_n)-f(x_n)}{y_n-x_n}=L$ then $f'(a)=L$

Let $f:I\rightarrow \mathbb{R}$ continuos in $a\in I$ interior point. If $$\lim \frac{f(y_n)-f(x_n)}{y_n-x_n}=L$$ for every $(x_n)$, $ (y_n)$ with $x_n<a<y_n$ and $\lim x_n=\lim y_n =a$ then ...
0
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0answers
14 views

How to bound error when approximating ODE

I have a question regarding how to bound the error, if one changes the "right hand side" of an ODE. For example, the equation of a simple pendulum in polar coordinates is something like ...
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2answers
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Why this function is uniformly continuous at $(a,b)$

I saw a theorem that say: "Let $f:\left(a.b\right)\rightarrow \mathbb{R}$ be continuous function. So, $f$ is uniformly continuous in $(a,b)\Leftrightarrow $ $\lim\limits_{x\to a^+}f(x)$ and ...
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Cauchy's sequence in sup-form

How to prove the below theorem ? Prove that the sequence $f_1$, $f_2$,... converges uniformly to a continuous function, $f\colon [0, 1] \to \mathbb{R}$, by showing that it is a Cauchy sequence in ...
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3answers
34 views

Continuous function simple inequality

I have the following simple question: if $f(x)$ is a continuous function and satisfies $f(x)\leq K$ for $x<a$ for some $K>0$ and $a$. Then, it is satisfies $f(a)\leq K$. I tried to find a ...
0
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1answer
25 views

Interchange limit of one variable with partial derivative of another variable

Consider a function $f(t,y)$ where $t,y \in \mathbb{R}$. It is given that $ f(t,y)$ is continuous but $\dfrac{\partial f(t,y)}{\partial t}$ suffers a jump at at $t= x$. (The function $f$ could ...
2
votes
3answers
41 views

existence of a unique continues function

Prove that there is exactly one continuous function $f\colon [0,\infty)\rightarrow \mathbb{R}$ that for every $x\geq0$ $f (x)=3f (2x)+e^x$ I've tried a lot, but All I know is: $f (0)=\frac{-1}{2}$ ...
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2answers
42 views

Show that the function $f(x, y)$ = $xy$ is continuous.

How do I show that $xy$ is continuous? I know that the product of two continuous functions is continuous but how do I show that $x$ is continuous and $y$ is continuous?
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1answer
16 views

Understanding and Proving Continuity of a Function in a Broad Sense

At the moment I am studying a chapter about Sobolev spaces in a book on partial differential equations authored by Evans. I have a question about continuity of a function in a broad sense. More ...
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2answers
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Continuous path in $\mathbb{R}^{n}$ passes through $D^{n}$ finitely many times.

I'm facing an algebraic topology exercise, and I only need to prove that in the title to finish it: Let $f: I \to \mathbb{R}^{n}$ a continuous path that passes through the unit sphere $D^{n} = \{ x ...
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0answers
28 views

Suppose $0 \lt a \lt 1$, but that $a\neq 1/n$. Find a function, continuous on $[0,1]$ and $f(0)=f(1)$, but $f(x)\neq f(x+a)$. [duplicate]

Suppose $0 \lt a \lt 1$, but that $a$ is not equal to $1/n$ for any natural number $n$. Find a function $f$ which is continuous on $[0,1]$ and which satisfies $f(0)=f(1)$, but which does not satisfy ...
4
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2answers
99 views

Let $f$ be a function such that every point of discontinuity is a removable discontinuity. Prove that $g(x)= \lim_{y\to x}f(y)$ is continuous.

Let $f$ be a function with the property that every point of discontinuity is a removable discontinuity, i.e., $\lim_{y \to x}f(y)$ exists for all $x$, but $f$ may be discontinuous at some (even ...
0
votes
2answers
23 views

Determine whether there exists a value of aϵℝ such that f(x, y, z) is continuous at the origin.

Consider the function $f:ℝ^3\toℝ$ defined by $f(x, y, z) = {(z/x) + y}$ if $x≠0$, $a$ if $x=0$ where $aϵℝ$ is some fixed constant. Determine whether there exists a value of $aϵℝ$ such that $f(x, ...
1
vote
1answer
42 views

How do I go about figuring out delta-epsilon proofs?

I'm going through Bert Mendelson's Introduction to Topology on my own. In fact, I've tried to go through it several times and I always get stuck somewhere. This time it's on limits. I think I ...
0
votes
0answers
16 views

Proving that the function $\rho$ which sends a lifting of a circle map to its rotation number is continuous.

Let $\mathcal{L}$ denote all circle maps of degree one with nondecreasing liftings (a map $f \in \mathcal{L}$ is of degree one if its lifting $F$ satisfies $F(x+1)=F(x)+1$) . I need to prove that if ...
1
vote
1answer
21 views

Lipschitz's continuity general question

Prove that lipschitz continuous (in its domain) function $f(x)$ defined on a bounded set, has bounded range. I do not know whether this is not directly implied by the definition, nevertheless I do ...
0
votes
1answer
29 views

Lipschitz continuity of $\frac{1}{x}$ and $x^2$

Show that function $f(x)=\frac{1}{x}$ fulfills lipschitz continuity on all rays $(\epsilon, + \infty), \epsilon>0$ whereas does not fulfill lipschitz continuity on ray $(0, +\infty)$ Show that ...
0
votes
1answer
18 views

Show that $m(\Gamma)=0$, where $\Gamma$ is a curve $y=f(x)$

Suppose $\Gamma$ is a curve $y=f(x)$ in $\mathbb{R}^2$, where $f$ is continuous. Show that $m(\Gamma)=0$. Hint: Cover $\Gamma$ by rectangles, using the uniform continuity of $f$. If the ...