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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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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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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37 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 ...
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1answer
32 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 ...
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1answer
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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| < ...
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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 ...
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2answers
40 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
24 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 ...
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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) = ...
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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 ...
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1answer
97 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 ...
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0answers
18 views

Show that if f is differentiable at $x_0$, then it is continuous at $x_0$. (Weierstrass-Caratheodory formulation)

this is an argument for a question which I am unsure whether it is sufficient or not. We are asked to try show the continuity at $x_0$ given that $f$ is differentiable at $x_0$. My argument goes as ...
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1answer
58 views

Showing this function is continuous $ f:(x,y)\mapsto x^2+y^2$

I have the following function: $$f:\Bbb R^2 \to \Bbb R,\quad f:(x,y)\mapsto x^2+y^2$$ I want to show that this function is continuous by showing that $f^{-1}((a,b))$ is an open set. How do I ...
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68 views

Continuity of $\frac{1}{|x|}$ at $x= 0$

The function $|x|$ is continuous at zero. What can I say about the continuity of $\frac{1}{|x|}$ ? I have two counter arguments for it continuity. Please suggest what is right. The function is not ...
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84 views

Prove that $ne^{-na} \leq C e^{\frac{-na}{2}}$

How to Prove that for any $a > 0$ there exists $C \in R$ such that for all $n \geq 1$ $$ne^{-na} \leq C e^{-na/2}$$
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1answer
120 views

Function f is bounded on $[0, \infty)$

Let $f : [0, \infty) \to R$ be continuous such that $lim_{x \to +\infty} f(x) = 0$. How can I Prove that f is bounded on $[0, \infty)$. I know that condition for a function to be bounded is - ...
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1answer
25 views

Proving that there exists a horizontal chord with length $1/n$ for a continuous function $f: [0,1] \to \mathbb R$

Given a continuous function $f: [0,1] \to \mathbb R$ and that the chord which connects $A(0, f(0)), B(1, f(1))$ is horizontal then prove that there exists a horizontal chord $CD$ to the graph $C_f$ ...
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45 views

If $f: (a, b) \to \Bbb R$ is uniformly continuous, then $g: [a, b] \to \Bbb R$ is continuous

Fact 1: If $A$ is a subset of a metric space, $Y$ is a metric space, and $f: A \to Y$ is uniformly continuous, then if $\{x_n\}$ is Cauchy in $A$, then $\{f(x_n)\}$ is Cauchy in $Y$. Fact 2: ...
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compact sets and continuity

Let be $X\subset \mathbb{R^m}$, $K\subset \mathbb{R^n}$ compact, $f : X\times K \rightarrow \mathbb{R^p}$ continuous and $c\in \mathbb{R^p}$. Suppose that for every $x\in X$, there is a unique $y \in ...
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207 views

Continuity of sin x over rationals

I need a little help on the following question: For the function $f\colon[0, 2\pi]\to\mathbb R$ defined below, explain with proof, at which points of $c \in [0,2 \pi]$ $f$ is continuous or ...
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1answer
55 views

How to prove funciton continuity - hard one

Let $f : (-5; 5) \rightarrow \mathbb{R}$ be given as $f(x) = \sum_{n = 5}^{n = \infty}{\frac{1}{n^2-x^2}}$.How to prove that $f$ is continous? Is $f$ differentiable? If yes, how to compute $f'(0)$? ...
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1answer
22 views

Type of discontinuities in Dirichlet function

I notice three types of discontinuities "removable", "jump", "infinite" defined here Classification of Discontinuities involve limit. Then I am puzzled that what is the type of discontinuities in the ...
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28 views

Let $f$ be continuous on $I=[a,b]$ such that $f(a)<0$, $f(b)>0$, $W= \{x\in I: f(x)<0\}$, $w=\sup W$ Prove that $f(w)=0$

Let $f$ be continuous on $I=[a,b]$ such that $f(a)<0, f(b)>0, W= \{x\in I: f(x)<0\}$, $w=\sup W$ Prove that $f(w)=0$ [I can see that this is an alternate proof for the Location of roots ...
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1answer
17 views

Looking for a continuous, unit-norm vector field

I want to find a 2D vector field with three characteristics: it is continuous all the vectors are unit length vectors on the unit circle point to the origin Is this possible? I haven't been able ...
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1answer
71 views

Continuity of $f(z) = Log z$ , for $z$ complex, non-real $-\ln|z|$, for $z$ real

At what points in the complex plane is this function continuous (If there is any)? Would it be correct to conclude that $f$ is then continuous for all $z$ in the complex plane less the real ...
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How do I show that this function is continuous? [duplicate]

I'm interested in showing that $CX=\frac{I\times X}{\{1\}\times X}$ is contractible. I defined the d.r $F(s,[t,x])=[(1-s)t+s,x]$ and the only missing part for me is to show that it is continuous. How ...
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How do I show that the inversion mapping for linear transforms is continuous in the operator norm?

I'm working through some analysis textbooks on my own, so I don't want the full answer. I'm only looking for a hint on this problem. My question is related to this question, but the textbook I'm ...
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Prove non-existance of limit: $f(x,y) = \frac{xy\sin(\frac{x}{y})}{x^2 + |y|^3}$

I need to prove that $f(x,y) = \frac{xy^2\sin(\frac{x}{y})}{x^2 + |y|^3}$ does not tend to $0$ when $(x,y)$ approaches $(0,0)$. In order to do so, I would need to find some direction $\alpha$ such ...
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1answer
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If $f\circ g$ is continuous and $f$ is a local homeomorphism, then $g$ is continuous

Suppose $g:X\to Y$ and $f:Y\to Z$, and $f$ is a local homeomorphism, which is to say that for any $y\in Y$ there is a neighborhood $U$ of $y$ such that $f\restriction U$ is a homeomorphism from $U$ to ...
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1answer
32 views

Show that if $\lim_{x \to a}f(x)=L$ then $f$ is bounded near $a$.

The Problem: Show that if $\lim_{x \to a}f(x)=L$ then $f$ is bounded near $a$, i.e. there are constants $C,M > 0$ such that $\left|f(x)\right|<M$ for all $x$ such that $\left|x-a\right| < ...
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Continuity of $\arctan\big( \frac{ln(2-x)}{(x-2)}e^x \big)$

I have to examen the continuity of this function: with a an element of [0, +∞[ So far, I've found this: Using the basic algebraic functions I can rewrite f(x) as: ...
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245 views

If $f(A)\to A^{-1}$, prove that $f$ is continuous.

Let $f \colon GL_{n}(\mathbb{R})\to GL_{n}(\mathbb{R})$ be a function which maps $A\mapsto A^{-1}$. Prove that $f$ is continuous. $GL_{n}(\mathbb{R})=\det^{-1}(\mathbb{R}\setminus\{0\})$ is ...
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To prove that no such function can be continuous. [closed]

Suppose $f: [a,b] \to R$ is two to one. that is, for each $y$ in $R$, $f^{-1}({y})$ is empty or contains exactly two points. How to prove that no such function can be continuous.
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When is this function differentiable $g(x)=(a +|x|)^2 \cdot e^{(5-|x|)^2}$?

Given a function : $$g(x)=(a +|x|)^2 \cdot e^{(5-|x|)^2}$$ Find the values of $a$ for which the function is continuous in $\mathbb R$ and the values for which it is differentiable. The function ...
2
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1answer
79 views

Prove $f$ has a maximum on a continuous $f: [0, \infty) \to [0, \infty)$ if $\lim\limits_{x \to \infty} f(x) = 0$

Prove $f$ has a maximum on a continuous $f: [0, \infty) \to [0, \infty)$ if $\lim\limits_{x \to \infty} f(x) = 0$. So $\lim\limits_{x \to \infty} f(x) = 0$ means that for all $\epsilon > 0$ ...
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1answer
32 views

Why is T continuous?

Let $T\colon X\to X$, with $X=\left\{0,1,2\right\}^{\mathbb{Z}}$, desribe the following dynamics: 1 becomes 2 2 becomes 0 0 becomes 1 if at least one of its two neighbours is 1, otherwise it remains ...
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Closed subset of $\mathbb{R}^2$ induced by the graph of a function

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. I want to show that the set $$A := \{(x, y) \in \mathbb{R}^2|y ≤ f(x)\}$$ induced by $f$ is a closed subset of $\mathbb{R}^2$. Now ...
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Does extreme value theorem hold for neighborhoods near $ \pm \infty$

Given a continuous function $f: \mathbb R \to \mathbb R $ can we state that $f$ satisfies the conditions of the extreme value theorem at an interval $A=[x_0, x_0 + c], c \in \mathbb R$ as $x_0 \to + ...
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1answer
44 views

A characterization of continuity

A set $O \subset \mathbb R$ is open if for any $x \in O$ there exists $ \eta > 0$ such that $(x - \eta, x + \eta) \subset O$. How can it be proved that if $f : \mathbb R \to \mathbb R$ is ...
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48 views

continuity and differentiable equivalent to 0

This is how I have the proof for this set up. However now I'm not really sure that saying "f'(x)=(f(1)-0)/1=f(1). This is equivalent to f'(x)=f(b). Hence we can write that f'(x)≥f(x)." is the ...
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1answer
15 views

Is $t\mapsto\sum_{i=1}^n1_{(t_{i-1},t_i]}(t)$ with $0=t_0<\ldots<t_n$ left- or right-continuos?

Let $0=t_0<\ldots<t_n$ and $$f(t):=\sum_{i=1}^n1_{(t_{i-1},t_i]}(t)$$ I'm confused whether $f$ is left-continuous or right-continuous. How can we prove it? It seems to be easy, but I can't ...
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Example 5, Sec. 24 in Munkres' TOPOLOGY, 2nd ed: Is this map always continuous?

Let $(X, \Vert \cdot \Vert)$ be a given normed space that has elements other than the zero vector $\theta_X$. And let $T \colon X-\{\theta_X \} \to X$ be defined by $$T(x) \colon= \frac{1}{\Vert x ...
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1answer
47 views

How differentiable is the function $g(x) = \sum_n 2^{-n} f(x-r_n)$ where $f(x)=x^2 \sin\frac1{x}$?

This is an auxiliary enquiry (something like it may well be already discussed on MSE, but I haven't found it) resulting from a feeling of unease provoked by the question of this post. Taking the ...
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1answer
58 views

Can a function be differentiable while having a discontinuous derivative?

Recently I came across functions like $x^2\sin(1/x)$ and $x^3\sin(1/x)$ where the derivatives were discontinuous. Can there exist a function whose derivative is not conitnuous, and yet the function is ...
2
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1answer
25 views

On general topological spaces and $C(X, \mathbb R)$ , where for closed sets $A,B$ in $X$ , $I_A=I_B \implies A=B$

Let $X$ be a metric space and $C(X, \mathbb R)$ be the ring of all real valued continuous functions from $X$ . For $A \subseteq X$ , let us define $I_A :=\{f \in C(X, \mathbb R) : f(x)=0 , \forall x ...
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0answers
27 views

Construct a bijection

Suppose $\phi: X \rightarrow Y$ and $f:X \rightarrow \mathbb{R}, g:Y \rightarrow \mathbb{R}$ where $X, Y$ are metric spaces and $f, g$ are Baire-$1$ functions. Let $x_0$ be the only point of ...
2
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4answers
66 views

A formal proof that the function $ x \mapsto x^{2} $ is continuous at $ x = 4 $.

Problem: Show $f(x)=x^2 $ is continuous at $ x = 4$. That is to say, find delta such that: $ ∀ε>0$ $ ∃δ>0 $ such that $ |x-a|<δ ⇒ |f(x)-f(a)|<ε$ Where $a=4$, $f(x)=x^2$,and $f(a)=16$. ...
2
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1answer
43 views

The set of points of continuity of a real-valued function on a metric space is a $G_\delta$ set

Let $f$ be a real-valued function on a metric space $X$. Show that the set of points at which $f$ is continuous is the intersection of a countable collection of open sets. I know lots of other ...
2
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2answers
58 views

continuity in an interval?

Let $g : [a, b] → \mathbb{R}$ be a continuous function on $[a, b]$. Given any $n \in \mathbb{N}$ and $x_1, . . . , x_n ∈ [a, b]$, show that there exists $x_0 ∈ [a, b]$ such that $g(x_0) =(g(x_1) + · · ...