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) ...

learn more… | top users | synonyms (1)

1
vote
0answers
20 views

Continuous Functions with graphing [closed]

I was working on the following problem: Show $f(x)$ is a nowhere continuous function whose absolute value is everywhere continuous $$f(x) = \begin{cases}1 & x \in \mathbb{Q}\\ -1 & x \...
3
votes
3answers
275 views

Are derivatives always continuous? [duplicate]

I am assuming first off that the derivative exists everywhere on the real number line (or everywhere in whatever set you choose to work in if for some insane reason you drag complex numbers or ...
1
vote
1answer
62 views

Corollary of Tietze extension theorem

The Tietze extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $g:...
1
vote
0answers
81 views

Continuity of $f(x, y)=\frac{xy}{\sqrt{|x|} +y^2}$ at $(0,0)$

Assume that $f: \Bbb R^2 \rightarrow \Bbb R$ is defined by $f(0,0)=0$ and, for every $(x,y)\ne(0,0)$, $$f(x, y)={xy \over {\sqrt{|x|} + y^2}}.$$ I have to check whether the function is continuous ...
6
votes
2answers
110 views

$f$ non-constant on $\mathbb R$ such that for any metric $d$ on $\mathbb R$ , $f:(\mathbb R,d)\to (\mathbb R,d)$ is continuous , is $f$ identity?

Let $f:\mathbb R \to \mathbb R$ be a non-constant function such that for any metric $d$ on $\mathbb R$ , $f:(\mathbb R,d)\to (\mathbb R,d)$ is continuous , then is $f$ the identity function i.e. $f(x)=...
4
votes
1answer
59 views

To characterize uncountable sets on which there exists a metric which makes the space connected

For which uncountable sets $X$ is it true that there exist a metric $d$ on $X$ such that $(X,d)$ is connected ? [ The motivation for this question is : I wanted to characterize function $f : X \to X$...
1
vote
4answers
109 views

What am I supposed to prove here actually?

I got problems understanding the task: Prove that there are at least two different $x \in (0, 2)$ such that $x^3-x-\sqrt{x} +\frac{1}{2}= 0$. The given hint is to use the intermediate value ...
0
votes
1answer
12 views

An increasing function defined on a interval that is only continuous outside a countable set

Let $C$ be a countable subset of $(a,b)$. Then there is an increasing continuous function on $(a,b)$ that is continuous only on $(a,b)\setminus C$ This is an example from Royden's real analysis book....
1
vote
1answer
30 views

Continuity of $f_\alpha(x,y)=\frac{xy}{(x^2+y^2)^\alpha}, f_\alpha(0,0)=0$ in $(0,0).$

I've already proved that $f_\alpha$ is discontinuous if $\alpha\geq 1$. Now I want to prove (what I assume, but don't know) that $f_\alpha$ is continuous if $\alpha<1$. The definitions of ...
0
votes
1answer
25 views

Does function that maps bounded convex sets (minus straight line segments) to bounded convex sets must be continuous everywhere?

This question in the title came to my mind while I was sitting with my granny in front of my house maybe about half an hour ago. Although it looks innocent I do not know at the moment some simple ...
2
votes
3answers
60 views

$f:S^1 \to \mathbb R$ be continuous , is the set $\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ infinite ?

Let $f:S^1 \to \mathbb R$ be a continuous function , I know that $\exists y \in S^1 : f(y)=f(-y)$ where $y \ne -y $ (since $||y||=1$) , so that the set $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=...
3
votes
1answer
43 views

$f \in C [0,1]$ , $\lim _{x \to 0+} f(x)/x$ exists finitely , $\lim_{n \to \infty} n\Big(n \int_0^1 f(x^n)dx-\int_0^1 \dfrac {f(x)}x dx\Big)=$? [closed]

Let $f:[0,1] \to \mathbb R$ be a continuous function such that $\lim _{x \to 0+} \dfrac {f(x)}x$ exists finitely . Then does the limit $\lim_{n \to \infty} n\Big(n \int_0^1 f(x^n)dx-\int_0^1 \dfrac {f(...
0
votes
1answer
93 views

Existence of metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is everywhere discontinuous

Does there exist a metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is everywhere discontinuous ? It is motivated from this question which ...
1
vote
1answer
30 views

Prove that $f(x, y, z) = x^2+y^2+z^2+2x+2y+2z+3$ is a continuous function from $R^3$ to $R$. (i.e. show that $f^{-1}$ ((a, b)) is open in $R^3$ .)

I'll start by stating I have found a very similar question already posted, but that the solution the asker has accepted isn't helping me understand what I need to do Prove that a function is ...
1
vote
1answer
58 views

Prove that an arbitrary norm is continuous. Is my proof correct?

Let $f: \mathbb{F}^n\rightarrow \mathbb{R}$ be defined by $f(a_1,\cdots, a_n)=\|\sum a_jv_j\|$. Show $f$ is continuous on $\mathbb{F}^n$. 1. $\|\cdot\|$ is an arbitrary norm on $\mathcal{V}$. ...
1
vote
0answers
92 views

Continuous embeddings

Given the following exercise: We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ (the ...
1
vote
0answers
55 views

Continuity of the period of solutions of a second order ODE, with respect to their initial conditions

There is a second order ODE $$\ddot{x} + b(x) \dot{x}^2 + c(x) = 0$$ with continuous, and locally lipschitz coefficients b, c : $\mathbb{R}\to\mathbb{R}$. Assume the ODE has 2 partial periodic ...
1
vote
2answers
53 views

Continuous functions in Topologies

I'm having quite some difficulty finding continuous functions between topologies. Find a continuous function $f:\Bbb{R}_{cocountable} \rightarrow \Bbb{R}_{ususal}$ I'm not sure maybe something that ...
3
votes
2answers
83 views

Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is not continuous?

Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is not continuous?
1
vote
1answer
35 views

$\epsilon - \delta$ definition to prove that this graph is not continuous at $a$

It is a fundamental problem, and there are some related problem asked before: 1. $\epsilon - \delta$ definition to prove that f is a continuous function. 2. How to show that $f(x)=x^2$ is continuous ...
1
vote
0answers
121 views

Problem regarding continuous embeddings [duplicate]

Given the following exercise: We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \{x \in (\mathbb R_+) : f(x) \neq 0\} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ and for all $f \in C^...
0
votes
1answer
59 views

Absolute continuity and continuity

Suppose we have a measure $\mu$ on $(a,b]$ such that $\mu(a,b]=F(b)-F(a)$ where $F$ is non-decreasing, continuous function from the right, Definition: A function $F$ is said to be absolutely ...
1
vote
2answers
30 views

Continuous functions in a topology

The projection function $f : R_{usual}^2 \rightarrow R_{usual}$ given by f(x, y) = x is continuous. Can someone please provide a proof to this, in general when you need to show a function is ...
0
votes
1answer
44 views

Urysohn's extension theorem

Currently I am working my way through Ernest Michael's first article on continuous selections. Here, Urysohn's extension theorem is stated as follows: For a $T_1$-space, the following properties ...
3
votes
2answers
44 views

Compact Sets of $(X,d)$ with discrete metric

Let $X \neq \emptyset$. Define the discrete metric on $X$ with: $ d(x,y)=\left\{\begin{array}{ll} 1, & x \neq y \\ 0, & x=y\end{array}\right.$ (a) Ascertain the compact ...
0
votes
2answers
48 views

Show that $f(x) = \inf\{d(a,x) : a \in A \}$ is continuous

Let $A$ be a non-empty set in a metric space $(X,d)$. Define $f: X \to \mathbb{R}$ by $f(x) = \inf \{d(a,x) : a \in A \}$. Prove that $f$ is continuous. If $f$ is continuous, then $\forall \epsilon &...
0
votes
0answers
17 views

Showing $(C[0,1],d_{\infty})$ is connected

Prove that the metric space $(C[0,1], d_{\infty})$ is connected. Is it path connected? I know how to typically show that a set is connected, but to show $C[0,1]$ is connected is currently escaping ...
1
vote
1answer
34 views

Holder continuous but not Lipschitz

Is there a function that is Holder continuous but not Lipschitz continuous?
1
vote
2answers
67 views

Showing the continuity of $d(x,f(x))$

Assume that $(X,d)$ is compact, and that $f: X \to X$ is continuous. Show that the function $g(x) = d(x,f(x))$ is continuous and has a minimum point. Consider the function $g(x) = d(x,f(x))$. If $g$ ...
1
vote
3answers
72 views

How is the function $f: \mathbb{Z} \to \mathbb{R}$ continuous?

Where $\mathbb{Z}$ is the set of integers and $\mathbb{R}$ the set of real numbers. In a question in a problem sheet, it said this statement was correct, however I do not understand how. You ...
1
vote
4answers
58 views

The continuity of function's restrictions implies the continuity of function.

Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove ...
1
vote
2answers
52 views

Differentiability and continuity of $f(x)=x^2+\frac{x^2}{1+x^2}+\frac{x^2}{(1+x^2)^2}+\ldots \to \infty$

If $$f(x)=x^2+\frac{x^2}{1+x^2}+\frac{x^2}{(1+x^2)^2}+\ldots \to \infty \,,$$ then choose the correct option: $(A)$ lim($x \to 0$) $f(x)$ does not exist. $(B)$ lim($x \to 0$) $f(x)$ exist but $f(x)$...
2
votes
2answers
88 views

Prove that if $\lim _{x\to \infty } f(x)$,then $\lim_{x\to \infty} f(x)=0$

Let $f:\Bbb R\to \Bbb R$ be a continuous function such that $\int _0^\infty f(x)\text{dx}$ exists. Prove that if $\lim _{x\to \infty } f(x)$,then $\lim_{x\to \infty} f(x)=0$ If $f$ is ...
0
votes
1answer
34 views

show that the mapping $f: (\mathbb{R},\ell_1) → (\mathbb{R},\ell_2)$ is continuous

show that the mapping $f: (\mathbb{R},\ell_1) → (\mathbb{R},\ell_2)$ is continuous. $f(x) = 0, x=-3$ and $\sqrt 3, x\neq-3$ $ł_1=\{U\subset \mathbb{R}: U=\emptyset \lor-3\in U \}$ $ł_2=\{U\subset \...
3
votes
3answers
127 views

Continuity of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$? [duplicate]

Suppose a function $f$ is defined as follows: $$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$ Is this function ...
0
votes
0answers
15 views

show that Lb(a, y) := max{1−ay−by,0}, (a, y) ∈ R×{−1,1}, b ∈ R, is continuouse whit respect to first variable

Show that: $$L_b(a, y) = \max\{1−ay−by,0\},\;\; (a, y) \in\mathbb{R}×\{−1,1\}, b\in\mathbb{R}$$ is continuous whit respect to first variable.
2
votes
1answer
46 views
1
vote
1answer
40 views

$f \in C[a,b]$ , $f''$ exists in $(a,b)$ ; $\exists t \in (0,1) : f(ta+(1-t)b)=tf(a)+(1-t)f(b)$ ; then $\exists c \in (a,b)$ such that $f''(c)=0$? [closed]

Let $f:[a,b] \to \mathbb R$ be a continuous function , twice differentiable in $(a,b)$ , such that $\exists t \in (0,1)$ such that $f(ta+(1-t)b)=tf(a)+(1-t)f(b)$ ; then is it true that $\exists c \...
0
votes
1answer
33 views

Find $f(2^2)$ in given condition [duplicate]

Let $f(x)$ be a continuous function in $[1,3]$ defined for all $x$ belonging to $ R $. If $f(x)$ take rational values for all $x$ belonging to R and $f(2)=198$ then $f(2^2)$
0
votes
2answers
43 views

How to prove that a function is continuous

Generally a function is shown continuous by directly taking left hand or right hand limit.But sometimes the same can be shown continuous by letting h tend to zero.What is the difference,would somebody ...
2
votes
2answers
17 views

What is the definition of a path along a multivariable function?

I'm taking a class equivalent of Calculus III, and we saw how to prove continuity of a multivariable function. Recently we looked at the following example: \begin{align} f(x,y) = \begin{cases} \frac{...
2
votes
2answers
151 views

Not continuous then limit does not exist? [closed]

Does a function is not continuous in an interval implies its limit does not exist at any point in the interval? EDIT: the function is not continuous at every point in the interval. Sorry.
2
votes
4answers
60 views

periodic continous function $f$

I have a function $f: \mathbb{R} \longrightarrow \mathbb{R}$ that is continous with $f(x)=f(x+2)$ for all real numbers $x \in \mathbb{R}$. So, I have to show that an $a \in \mathbb{R}$ exists that $f(...
0
votes
1answer
30 views

Proving that a plane reaches a certain velocity at least two times during a flight

I am asked to prove the following: A plane initiates its departure at 2pm. The distance it will travel is $2500~\text{mi}$. The plane arrives at its destiny at 7:30pm. Prove that, at least two times ...
2
votes
2answers
79 views

Is $(x^2+y^2+z^2) \ \sin \frac{1}{\sqrt{x^2+y^2+z^2}}$ Differentiable in $(0,0,0)$?

$$f(x,y,z)=\begin{cases} (x^2+y^2+z^2) \ \sin \frac{1}{\sqrt{x^2+y^2+z^2}} \qquad (x,y,z) \ne (0,0,0) \\ \\ 0 \qquad (x,y,z)=(0,0,0) \end{cases} $$ At first, I study the continuity in the origin. I ...
1
vote
1answer
25 views

Proving continuity of this Dirichlet function

I'm prepping for a final and I have a question regarding the following Dirichlet function. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined by : $$f(x) = \begin{cases} x^2 & x \in ...
1
vote
1answer
24 views

Prove absolute continuity without Banach-Zarecki

Let $f$ be a real-valued continuous function of bounded variation on $[a,b]$. Suppose $f$ is absolutely continuous on $[a+\eta,b]$ for every $\eta\in(0,b-a)$. Show that $f$ is absolutely continuous on ...
0
votes
1answer
19 views

Are continuous product projections always split?

Are continuous product projections always split? What's an example of a product projection without a continuous right inverse?
5
votes
1answer
78 views

For every interval $I$ in $\mathbb R$ , there exists a continuous surjection from $I \setminus \mathbb Q$ to $I \cap \mathbb Q$?

Is it true that for every interval (not singleton ) $I$ in $\mathbb R$ , there exists a continuous surjection $f : I \setminus \mathbb Q \to I \cap \mathbb Q$ ?
1
vote
1answer
53 views

Continuity of a Single Point

My problem is :Find the points at which the the mentioned function is continuous $$f(x) = \begin{cases} x & \text{if $x$ is a Rational Number} \\ -x & \text{if $x$ is not a ...