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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Is this condition on continuity extraneous or troublesome?

I was trying to motivate the use of open sets for defining continuity (as in topology or metric spaces). I came to formulate the following definition of continuity for a function $f: X \rightarrow ...
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1answer
17 views

find the Classification of discontinuities of a function

In my assignment I have to find the Classification of discontinuities of the following function: $$f(x)=\frac{\sin^2(x)}{x|x(\pi-x)|}$$ I wanted to start with the value $x=0$ because the function ...
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2answers
29 views

The function is not continuous

$$C([a,b])=\{ f: [a,b] \to \mathbb{R} \text{ continuous} \}$$ $C([a,b])$ is a linear space. For $f \in C([a,b])$ we define $\|f\|_{\infty}:= \sup_{x \in [a,b]} |f(x)|$ and easily it can be shown ...
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4answers
59 views

Show that if $f$ is continuous at $a$ and $f(a)≠0$ then $f$ is nonzero in an open ball around $a$.

Here is the question I'm dealing with: Let $U$ be an open set of $\mathbb{R}^{n}$, $f:U\rightarrow\mathbb{R}^{n}$ a function and $a\in U$ a given point. Show that if $f$ is continuous at $a$ and ...
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1answer
63 views

If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Let $X$ be a metric space such that every continuous function $f:X \to \mathbb R$ is uniformly continuous ( here $\mathbb R$ is equipped with the standard euclidean metric ) , then is it true that for ...
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4answers
32 views

Continuous function does not map closed set to closed set

I have a question in my textbook ask me to use this function $f(x)=x^2/(1+x^2)$ to show that continuous function does not necessarily map a closed set to a closed set. But I can't find any example to ...
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5answers
309 views

How is this example not a homeomorphism?

I am a beginner in Topology. I was going through Munkres book where I came across this example. The mapping $[0,1)\to S^1$ (unit circle) is bijective and continuous, but $f^{-1}$ is not continuous. ...
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2answers
28 views

If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$

Let $f:(X,\tau_X)\to (Y,\tau_Y)$ Prove: If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$. Could someone verify the following proof? ...
2
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2answers
47 views

Antiderivative is continuous

The following comes from Bass' book on Real Analysis: (Here $dy$ is Lebesgue measure) Exercise 7.6 Suppose $f:\mathbb{R}\to\mathbb{R}$ is integrable, $a\in \mathbb{R}$, and we define ...
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1answer
28 views

Why is the identity function from $\Bbb R$ with the Euclidean metric to $\Bbb R$ with the discrete metric not continuous?

Using only the definition of sequential continuity, show an example that $f(x) = x: \Bbb R \to \Bbb R'$ is not continuous, where $\Bbb R'$ has the discrete topology. So the definition of ...
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0answers
31 views

Lower semicontinuity on a metric space

I'm trying to prove something about lower semicontinuity for a map on a metric space $(X,d)$. I will try to write here my idea of the proof, hope someone can approve or contest it. Def. Let $(X,d)$ ...
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1answer
37 views

How to prove a function is continuous on a compact set?

I´m struggleing with this problem: I know by theorems that inf(d(a,b)) exists if the real value function d is continuous on the set AxB. But how can I prove that d is continuous?
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1answer
23 views

nonempty disjoint closed subsets of $\mathbb{R}^{n}$ [on hold]

Let $A$ and $B$ be two nonempty disjoint closed subsets of $\mathbb{R}^{n}$. For $x\in{\mathbb{R}^{n}}$ let $f(x)={d(x,A)/{(d(x,A)+d(x,B))}}$ Show that $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is ...
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0answers
21 views

Sequence converges to x0 iff inverse of f is continuous at f(x0) [on hold]

Let $A$ and $B$ be two nonempty subsets of $\mathbb R^n$ and $f:A\to B$ be a bijection. Show that $f^{-1}$ is continuous at a point $y_0=f(x_0)$ iff whenever $\|f(x_n)-f(x_0)\|\to0$, the sequence ...
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3answers
74 views

How to show that $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function

(This is a homework problem) I am trying to show that the series $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function on $\mathbb{R}$. My idea was to show that the functions ...
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0answers
19 views

Topological Embedding Which is Neither Open nor Closed

I'm having trouble coming up with an example of an embedding which is neither open nor closed. My attempts have included trying to find such a map from $\mathbb{R}$ (given the usual Euclidean ...
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1answer
28 views

Showing a function is not continuous at any other points.

I am having trouble picking a number x in the interval below. I need help picking the right interval. I tried sketching it, but I think I have trouble understanding it. Can someone help me clarify ...
0
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2answers
27 views

Injective implies invertible? Injective and well-defined implies bijective?

I have two questions regarding functions regarding linear maps: (Let $X$ and $Y$ be to Banach spaces) If $T:X\rightarrow Y$ is injective, then $T^{-1}$ exists, right? If $T:X\rightarrow Y$ is ...
2
votes
1answer
25 views

Topological field - Proving continuity of inversion

Given a field $F$ and an absolute value $|\ |$ on $F$, define the distance $d(x,y)$ between two elements $x,y\in F$ by $$ d(x,y) = |x - y|. $$ I just worked through the proofs that $d$ defines a ...
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5answers
63 views

Show that $f(x) = 0$ for all $x \in \mathbb{R}$ [duplicate]

$f: \mathbb{R} \to \mathbb{R}$ is continuous with $f(x)=0$ for all $x \in \mathbb{Q}$. Show that $f(x) = 0$ for all $x \in \mathbb{R}$. Can anyone please point me in the right direction as to how ...
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2answers
22 views

Is a function that has Holder order bigger than one constant?

I see from the Wikipedia that if a function $f$ over $[a,b]$ is Holder continuous with order strictly bigger than one, i.e. $$|f(x) - f(y)| < K |x-y|^\alpha$$ for some constant $K$ and ...
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2answers
64 views

How $(f_n)$ converges uniformly on $[a, b]$

Let $(f_n)$ be defined and continuous on an interval $[a, b]$, and differentiable on $(a, b)$. Let $c \in [a, b]$. Assume that $(f_n(c))$ converges and that $(f'_n)$ converges uniformly on $(a, b)$. ...
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4answers
31 views

Epsilon-Delta Continuity proof (verification/help)

So, I am really bad at these problems, and I don't know why. Edit: The metric over $\Bbb R$ is assumed to be $|f(a,b)-f(x_1,x_2)|$ Problem statement: Define $f: \Bbb R^2 \rightarrow \Bbb R$ by ...
2
votes
1answer
28 views

continuous extension problem

I was practicing on some exercises because I have quiz tomorrow, and I got stuck at this exercise, so I wish that someone would help me. here is the exercise. Given $$g(x) = \frac{x^2 -16}{x-4}$$ ...
0
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0answers
10 views

Is there any variation of Lipschitz continuity, where one can bound difference between value of 2 functions which act on different space?

Lipschitz continuity can be used to bound the difference between value of a function at two different points. Is there any variation of this, where one can bound difference between value of 2 ...
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3answers
17 views

Proving continuity with two different metrics

Problem statement: Let $X$ be the set of all continuous functions $f:[a,b]\rightarrow \Bbb R$, and define the metric $d^*(f,g)$ on $X$ by $$d^*(f,g) = \int_{a}^{b} |f(t) - g(t)|dt$$ Now, for each ...
3
votes
2answers
40 views

Proving the distance between two points is 1 on a continuous function

this is a tough question. We suppose $f$ is continuous on $[0,2]$ and $f(0) = f(2)$. We want to prove $\exists$ $x, y \in [0,2]$ such that $ \lvert y-x \rvert = 1 $ and $ f(x) = f(y)$. My attempt ...
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2answers
50 views

Let $n$ be an odd natural number , to find a continuous real valued function on $\mathbb R$ which takes every value exactly $n$ times

Let $n$ be an odd natural number . We know $\mathbb R = \cup_{k \in \mathbb Z} [nk\pi , n(k+1)\pi]$ . So for every $k \in \mathbb Z$ , define $h(x):=2k+1-(-1)^k \cos x , \forall x \in [nk\pi , ...
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1answer
28 views

How does differentiability affect the extremum of a function?

I have this function $$f(x)= \begin{cases} (x+1)^3 & -2< x\le-1\\ x^{2/3}-1 &-1<x\le1\\ -(x-1)^2 &1<x<2 \end{cases}$$ I'm supposed to find the total number of maxima and ...
6
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2answers
135 views

Prove $xe^x =2$ for some $x \in (0,1)$

We are trying to prove $xe^x =2$ for some $x \in (0,1)$. I know for certain that if this question were asking to prove the equality for some $x$ in the closed interval $[0,1]$ then I could apply ...
0
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1answer
42 views

Prove $g(x) = x^3$ is continuous at $x_0$ arbitrary

We are proving $g(x) = x^3$ is continuous at $x_0$ arbitrary. My attempt: For all $\epsilon \gt 0$, there exists $\delta \gt 0$ such that for an arbitrary point $x_0$ and $ \lvert x - x_0 \rvert ...
2
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1answer
56 views

Prove $f(x) =x^2$ is continuous at $x_0=2$ using the $\epsilon$-$\delta$ definition

We want to prove $f(x) =x^2$ is continuous at $x_0=2$ using the $\epsilon$-$\delta$ definition. My attempt: We want the function $f$ to satisfy the definition of continuity, meaning : For all ...
2
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1answer
56 views

Critique my proof? Proving f(x) is continuous at x = 0 for a defined function.

We have a problem to solve: Let $f(x)$ = \begin{cases} 0, & \text{when $x = 0$} \\ x\sin(\frac1x), & \text{when $x\not=0$} \end{cases} Prove $f(x)$ is continuous at $x=0$. My attempt: ...
4
votes
1answer
22 views

Image of a bounded sequence by a convex continuous function in a Banach space

Let $(X, \Vert \cdot \Vert)$ be a Banach space, and $f : X \longrightarrow \mathbb{R}$ a convex function, continuous for the norm topology. Suppose that $x_n$ is a sequence which weakly converges to ...
1
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1answer
32 views

Showing the inverse of an open and continuous function is also open

Say we have an open set, defined as $ O ⊂ \mathbb{R}$, and $\forall \space x ∈ O \space \exists \space η > 0$ such that $(x − η, x + η) ⊂ O$. I want to prove that if $f : \mathbb{R} → \mathbb{R}$ ...
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1answer
46 views

Critique My PROOF! : Proving a function is continuous at only one point.

I'm working on some problems from my book. Could you guys critique my logic? Here is a particular one: Let $h(x) = x$ for rational numbers x and $h(x) = 0$ for irrational numbers. Show the function ...
0
votes
1answer
37 views

Show that the functional is continuous everywhere in $V$

Let $J: V \to \mathbb{R}$ be a linear functional and $V$ a linear space with norm. Show that if $J$ is continuous on $0 \in V$ then $J$ is continuous everywhere in $V$. That's what I have tried: ...
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52 views

Showing that a function is not continuous

$$f(x)= \left\{ \begin{array}{} x, &x\in \mathbb{Q} \\ 0, &x\in \mathbb{R} \setminus\mathbb{Q} \end{array} \right.$$ Show that f is not continuous ...
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3answers
61 views

When is a continuous function differentiable? [duplicate]

I have been doing a lot of problems regarding calculus. An utmost basic question I stumble upon is "when is a continuous function differentiable?" (irrespective of whether its in an open or closed ...
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16 views

Help with improper integral (maybe hyperreal function).

I need proof: $f: [1, \infty) \to \mathbb{F}$ continuous, proof: $\displaystyle\int_{a}^{\to\infty}f(ax) dx$ is convergent, (eventually to $\infty$), $\forall a\geq 1$ and ...
0
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1answer
36 views

Uniform continuity of $\sqrt{x^2+x}$

I have to say that I know the definition. I've tried to use is in practical way, but I still don't know how to do that and I don't truly understand that topic. Please show me step by step solution to ...
2
votes
1answer
30 views

almost continuous function

Here, definition $2$, we have a function $f:X \rightarrow Y$ is almost continuous at $x \in X$ in the sense of Husain iff for any open set $V \subset Y$ containing $f(x)$, we have the closure of ...
2
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1answer
43 views

A computation problem from Real Analysis

The problem states: Find a $\delta > 0$ so that $|x-2| < \delta$ implies that (a) $|x^2 + x - 6| < 1$ (b) $|x^2 + x - 6| < \frac{1}{n}$ for a given integer n (c) $|x^2 + x - 6| < ...
3
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2answers
45 views

Continuity vs. Mapping open sets to open sets?

I have a question and I have no idea how to solve this: One problem in my Real Analysis text book says: Show that if $\ell$ is a nonzero linear functional on a normed vector space not necessarily ...
2
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2answers
37 views

$\inf_{x\in[a,b]}f(x)=\inf_{x\in[a,b]\cap\mathbb{Q}}f(x)$ for a continuos function $f:[a,b]\to\mathbb{R}$

Let $f:[a,b]\to\mathbb{R}$ be continuos. I'm sure it's not hard, but I'm unsure what exactly we need to do to prove $$\inf_{x\in[a,b]}f(x)=\inf_{x\in[a,b]\cap\mathbb{Q}}f(x)$$
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1answer
42 views

Show that $a\to f(a)$ from $A$ to $S$ is continuous.

I am reading "Continuity" in Metric Spaces Suppose $S\subset \mathbb R$ is a closed set. Suppose $A\subset \mathbb R$ has the property that for every $a\in A$ there is a unique nearest point $f(a)$ ...
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1answer
19 views

A continuously differentiable bijection implies its inverse is Lipschitz continuous

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable bijection. Does this imply that $f^{-1}$ is Lipschitz continuous? (of course, not globally, take for instance $f(x)=x^3$) If ...
0
votes
1answer
37 views

Prove that if an even function $f(x)$ is $C^2$, $f(|x|)$ is also $C^2$ [duplicate]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a $C^2(\mathbb{R})$ function which is also even (ie, $f(-x) = f(x)$). Prove that the function $F: \mathbb{R^n} \rightarrow \mathbb{R}$ defined by $F(x) = ...
2
votes
2answers
36 views

Axiom of choice : continuous function and uniformly continuous

How I proof that every continuous function f in [0,1] is uniformly continuous, without axiom o choice? I took this from the book Axiom of Choice from Horst Herrilich He had a observation that ...
0
votes
1answer
54 views

Prove that the series is continuous and differentiable [closed]

How to prove that the series $\sum e^{-nx+\cos(nx)}$ is defined, continuous and differentiable (with a continuous derivative) on $(a, \infty)$ for any $a > 0$. I am good with continuity part. But ...