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)

2
votes
1answer
20 views

How to show the logical equivalence of the following two definitions of continuity in a topological space?

Definition 1. Let $(X,\mathfrak{T}_X)$ and $(Y,\mathfrak{T}_Y)$ be two topological spaces and $f:X\to Y$. Then we will say that $f$ is continuous at $x\in X$ if for all $\{f(x)\}\subseteq V\in ...
0
votes
0answers
13 views

Describe the characteristics of this function (where it is defined, continuous, diff, twice diff).

Consider the function $f(x)=\sum\limits_{k=1}^{\infty}\frac{\sin(x/k)}{k}$ Where is $f$ defined? Continuous? Differentiable? Twice-differentiable? My thoughts so far: Initially I thought that due ...
1
vote
1answer
49 views

Is $f(x) = \sqrt[3]{x}$ continuous on $ [0,\infty)$ and is it uniformly continuous on $[0,\infty)$

Note, there is a similar question here: Show $\sqrt[3]{x}$ is or isn't uniformly continuous. I am asking my question anyway because the one here does not ask about the Sequential Criterion for ...
3
votes
2answers
40 views

Why can we 'choose' continuity points?

Let $F$ and $F_n$ be distribution functions with $\lim_n F_n(x)=F(x)$ for all continuity points $x$ of $F$. In a proof there is the following part: Block quote [...] choose the finite points ...
1
vote
1answer
20 views

Continuity of the Box-Cox transform at λ = 0: Why is it the log?

The Box-Cox power transform frequently used in statistical analysis takes the value (x^λ -1) /λ for λ not equal to zero, and ln(x) for λ=0. I would like to see a demonstration, that need not be a ...
6
votes
4answers
277 views

Provide examples or explain why it is impossible

a) A continuous function defined on an open interval with range equal to a closed interval. My example: $f(x)=\frac{1}{2}\sin(4\pi x)+\frac{1}{2}$ on $(0,1)$ to $[0,1]$. Note: I am not considering ...
0
votes
0answers
11 views

Can a polynomial form any one to one and continuous graph?

Hello I was wondering if it was possible to right any one to one and continuous graph as a polynomial with real co-efficient s. If this is not so why? It seems like it would possible to just write a ...
0
votes
1answer
31 views

continuous of minimum

Let $\Omega$ be an closed bounded and connected domain in $R^n$ and $h(x,t)$ is continuous in $\Omega\times [0,T]$.Let $$ H(t)=\min_{x\in\Omega} h(x,t) $$ How to prove $H(t)$ is continuous ? What I ...
0
votes
2answers
35 views

Let $g$ be continuous on an interval $A$ and let $F$ be the set of points where $g$ fails to be injective,

That is, $F = \{x \in A: f(x)=f(y)$ for some $y \neq x $ and $y \in A \}$ Show that $F$ is either empty or uncountable. Case (i) $F$ is empty: I can show this with a function that is monotonic and ...
1
vote
1answer
47 views

Let $f: [0,1] \rightarrow \mathbb{R} $ be continuous with $f(0) = f(1)$ *note, there is a part b* [duplicate]

(a) Show that there must exist $x,y \in [0,1] $ satisfying $|x-y| = \frac{1} {2}$ and $f(x) = f(y)$ I can start by defining a function $g(x) = f(x + \frac{1} {2}) - f(x)$ to guarantee an $x,y$ so ...
2
votes
1answer
36 views

prove that $\exists\ \epsilon>0$ such that $\forall x\in [0,1] : f(x)>x+\epsilon$

the question itself: Let $f$ be a continuous function in the close interval $[0,1]$ which upholds the rule: $\forall x\in [0,1] : f(x)>x$. prove that $\exists\ \epsilon>0$ such that $\forall ...
0
votes
1answer
31 views

Show that a differentiable function $f:\mathbb{R} \to \mathbb{R}$ has a global max in $a$ if $a$ is its local max

My task is this: Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function and assume that the only stationary point $f$ has is a local max in the point $A = (a,f(a))$. Show that $A$ must be a ...
1
vote
1answer
11 views

Multivariate Non-Differentiability

This example says that "continuous partial derivatives imply differentiability but not vice-versa". Based on transposition logic, I would then assume that if a multivariate function has discontinuous ...
1
vote
2answers
53 views

Optimization with a Probability

Imagine two points in $ℝ^2$ at $(-1, 0)$ and $(1, 0)$. You would like to walk from one point to the next in the shortest distance possible. However, there is a line segment coming from the origin to a ...
0
votes
1answer
18 views

Lipschitz continuity of continuously differentiable function

Is it true that a continuously differentiable function in a Banach space $X$ is locally lipschitz in $X$?
1
vote
2answers
41 views

Differential Equation - Where does the solution end?

I was asked to solve the differential equation $y'+\frac{y}{x+1}=\frac{2y-1}{x}$, given the starting point y(0.5)=5/6. The equation meets the criteria for Existence and Uniqueness for every x>0 (as y' ...
2
votes
2answers
32 views

Let E be a bounded

Let $E$ be a bounded subset of $\mathbb{R}$, & let $S$ = sup($E$) be the least upper bound of $E$. $S$ is also a real number. Show that $S$ is an adherent point of $E$, & is also an adherent ...
2
votes
3answers
21 views

Number of real solutions of a cubic equation without using derivatives

The problem is to find the number of real solutions of a cubic equation. This exercise is in a book, in the chapter about functions, limits and continuity. This chapter is before the chapter about ...
0
votes
2answers
40 views

continuous function in a topological space

It is known that if $f, g$ are continuous functions then $f+g$ is also continuous. I want to know how to prove it in topological language, thst is, $f$ is continuous if for any $x$ and any open ...
0
votes
1answer
30 views

Proof Validation: of f(0)=0 given differentiable…

Let $f$ be a differentiable function on an interval $A$ containing $0$, and assume $(x_n)$ is a sequence in $A$ with $(x_n)$ converging to $0$ and $x_n\neq0$ $\forall n\epsilon\mathbb{N}$. Want to ...
1
vote
2answers
29 views

Homeomorphism from real interval to an arc of a circle

I haven't seen this question anywhere, surprisingly. In a proof of some theorem, my lecture note abruptly states the above. That Since there is a homeomorphism of any real interval and the arc of ...
2
votes
3answers
161 views

Continuous functions in the indiscrete topology?

Slight curiosity. I've learned not to question too much in topology and basically, acquiesce. In a sense, thinking hard or trying to be smart in this area of study is a suicide mission for newbies. ...
1
vote
1answer
56 views

Find 2 continuous functions $F$ and $G$ defined on $[a;b]$, such that $F'(x) = G'(x)$, but $F(x) - G(x) \neq \text{const}$

The problem: Find 2 continuous functions $F$ and $G$ defined on $[a;b]$, such that for every $[\alpha;\beta] \subset [a;b]$ there exists an interval $[\alpha';\beta'] \subset [\alpha;\beta]$, where ...
2
votes
2answers
70 views

continuous (on 3, 4 and 5) f is constant, if $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is continuous on 3, 4 and 5, such that $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$. Show that f is constant. I don't know what to do ...
0
votes
3answers
37 views

An increasing smooth map $f:(0,1)\rightarrow(0,1)$ which does not extend to any smooth function on a larger domain

Although I'm not sure it's related, I have found a smooth map $f:(0,1)\rightarrow(0,1)$ which does not extend to any continuous function on a larger domain, namely ...
3
votes
1answer
44 views

Explicit functions evaluated

(a) Defined $f$ by $f(y):=\int_0^\infty\frac{xy}{(x^4+y^4)^{3/4}}dx$. Prove $f(y)$ is defined (i.e integral exists) for every $y\in\mathbb{R}$. (b)Prove that actually $f(y)=c\operatorname{sign} y$ ...
1
vote
1answer
58 views

Is $f(x)=\sum_{n=2}^{\infty} \frac{1}{n\ln(n)^x}$ continuous on $(1,\infty)$?

Is $f(x)=\sum_{n=2}^{\infty} \frac{1}{n\ln(n)^x}$ continuous on $(1,\infty)$? I have proven that the infinite series converges on $(1,\infty)$. I want to use the Weierstrass M-test to prove this ...
4
votes
1answer
61 views

Continuous map in $\mathbb{R}^2$ has a (scaled) fixed point

Let $\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a continuous map. How do I prove that there exist $a>0$ and $x\in\mathbb{R}^2$ such that $\phi(x)=ax$? What I know: I thought maybe this can ...
0
votes
0answers
11 views

Prove using darboux sums that every continuous $ f : \mathbb{R}^n \rightarrow \mathbb{R}$ with bounded support is integrable.

Prove using darboux sums that every continuous $ f : \mathbb{R}^n \rightarrow \mathbb{R}$ with bounded support is integrable. My motivation is to try and use a step-function and approximate it ...
1
vote
1answer
64 views

Question about a continuous periodic function [closed]

Consider the continuous and periodic function $f:\mathbb R \rightarrow \mathbb R$ with period $T > 0$ so that $f(x)=f(x+T)$ for any $x$. Question: Prove that there exists a $c$ such that ...
0
votes
0answers
17 views

If $f$ is continuous in $[0,1]$

then $\lim_{n\rightarrow \infty} \Sigma_{j=0}^{[\frac{n}{2}]} \frac{1}{n} f(\frac{j}{n})$, ( where $[y]$ is the largest integer less than or equal to $y$ ? Since $f$ is continuous in $[0,1]$, so it ...
-1
votes
2answers
32 views

Show that there exist $x_{0}\in[0,1]$ such that $f(x_{0})=g(x_{0})$ [closed]

Let $f,g:[0,1]\rightarrow[0,\infty)$ be continuous such that $\smash{\displaystyle\max_{x \in [0,1]}} f(x) = \smash{\displaystyle\max_{x \in [0,1]}} g(x)$. Show that there exist $x_{0}\in[0,1]$ ...
0
votes
0answers
46 views

Show that $S^1$ acts on $S^3$

$S^3=\{(z_1, z_2) \in \mathbb{C^2} \mid |z_1|^2 + |z_2|^2 = 1 \}$ Show that $S^1$ acts on $S^3$ by $z \cdot (z_1, z_2)=(zz_1, zz_2)$ An action of a topological group $G$ on a topological ...
0
votes
1answer
28 views

Does continuity in $(X, d_X)$ imply continuity in $(Y, d_Y)$ when $(X, d_X) \simeq (Y, d_Y)$?

I want to check if my intuition about continuity is correct. Suppose $(X, d_X)$ and $(Y, d_Y)$ are two metric spaces that are isometrically isomorphic, i.e., there is an isomorphism $h : X \to Y$ ...
0
votes
3answers
34 views

Proving a Function is continuous on an interval.

For the function $f(x) = \frac {1}{\sqrt{x}}$ Show the function is continuous on (0, $\infty$) How do I approach/do this question?
-1
votes
2answers
59 views

How to calculate Hyperoperators with reals?

In the Chinese wiki page Hx(3;3) = 3[x]3 is calculated somehow: https://zh.wikipedia.org/wiki/File:Hyperoperation_3_and_3_with_real_number.svg How can they do it? What is the method?
1
vote
1answer
30 views

Let $f(x,y) = \begin{cases} 1, & \textrm{if } xy = 0 \\ xy, & \textrm{if } xy \neq 0 \end{cases}$

Then (A) $f$ is continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ exists (B) $f$ is not continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ exists (C) $f$ is continuous at ...
1
vote
0answers
19 views

Provide an example of each or explain why the request is impossible.

(a) Two functions $f(x)$ and $g(x)$, neither of which are continuous at $0$ but $f(x)+g(x)$ and $f(x)g(x)$ are both continuous at $0$ I said possible and let $f(x) = \{0: x<0, 1: x \geq 0 \}$, ...
0
votes
0answers
40 views

I do not understand the last process of proving that $f$ is continuous iff $f^{-1}(G)$ is open.

The problem is: Let $f$ be a finite function on $\mathbb{R}^n$. show that $f$ is continuous on $\mathbb{R}^n$ if and only if $f^{-1}(G)$ is open for every open $G$ in $\mathbb{R}^1$, or if and ...
2
votes
3answers
40 views

What is the logic underlying this proof?

Proposition: A metric space $X$ is connected if, and only if, every continuous function $f:X\to (\{0,1\},d_D)$ is a constant function, where $d_D$ is the discrete metric on the set $\{0,1\}$. ...
1
vote
1answer
32 views

Prove that if $x\mapsto -x$ is continuous then $\sigma$ is the discrete topology.

Let $\tau $ be the topology on $\Bbb R$ for which the intervals $[a,b)$ form a base.Let $\sigma$ be a topology on $\Bbb R$ such that $\sigma \supseteq \tau. $ Prove that if $x\mapsto -x$ is ...
2
votes
1answer
28 views

show that continuous functions on $\mathbb{R}$ are measurable

I am trying to show this using the theorem: A function $f: \Omega \to \mathbb{R}$ is measurable if and only if $f^{-1}(E) \in \mathcal{F}$ for all borel sets $E$. The proof to show a continuous ...
2
votes
1answer
44 views

Are $\lim_{h\to0}f(a+h)=f(a)$ and $\lim_{h\to0}f(x+h)=f(x)$ the same?

An exercise I came across in my calculus text is as follows: Prove that $f$ is continuous at $a$ if and only if $$\lim_{h\to0}f(a+h)=f(a)\tag{1}.$$ Now, I saw a proof of the Product Rule ...
5
votes
1answer
41 views

Let $f: [0, 1] \to \mathbb{R}$ s.t $f(0)=f(1)=0$ then measure of $A = \{h \in [0, 1] \mid \exists x \text{ such that }f(x+h) =f(x)\} \geq 1/2$.

Let $f:[0,1]\to\mathbb R$ be a continuous function s.t. $f(0)=f(1)=0$. Let $$A = \{h \in [0, 1] \mid \exists x \text{ such that }f(x+h) =f(x)\}.$$ Show that set $A$ has Lebesgue measure $\geq 1/2$. ...
2
votes
2answers
27 views

Checking if “continuous” when $x$ is 1 and reaches 1

I have $$f(x) = x \left| x - 1 \right|$$ Here my given value for $x$ is 1 And I need to test if the function is "continuous" when $x$ is $1$ and also when reaching $$ f(1)$$ $$ \lim\limits_{x \to ...
1
vote
0answers
32 views

When $F(t)=\int_0^tf(s)ds$ is differentiable everywhere?

Let $f:\mathbb{R}\to \mathbb{R}$ be a function that is continuous almost everywhere. 1) Is the function $F(t)=\int_0^tf(s)ds$ differentiable everywhere ? 2) What is the "weakest" condition on $f$ ...
3
votes
1answer
53 views

Function that is second differential continuous

Let $f:[0,1]\rightarrow\mathbb{R}$ be a function whose second derivative $f''(x)$ is continuous on $[0,1]$. Suppose that f(0)=f(1)=0 and that $|f''(x)|<1$ for any $x\in [0,1]$. Then ...
1
vote
0answers
18 views

Continuous function rational for every point, Cantor function

For Cantor function (https://en.wikipedia.org/wiki/Cantor_function), in my sense it is rational on every point. But it is continuous on [0,1], then such a function must be constant. What is the ...
1
vote
0answers
29 views

Let $\alpha$ be a real number. Find the value of $\alpha$ for which the given function is continuous and differentiable.

Let $\alpha$ be a real number. Consider the function $$g(x)=(\alpha+|x|)^2e^{(5-|x|)^2}, \ \ \ -\infty<x<\infty $$ $(i)$ Determine the values of $\alpha$ for which $g$ is continuous at all $x$. ...
1
vote
2answers
48 views

Using the $\epsilon-\delta$ definition show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$

Using the $\epsilon-\delta$ definition show that $f(x) = \frac 1 {x^2}$ is a continuous function at any $x_0 = a, a > 0$ I have expressed in the form: $$lim_{x\to a}\frac1{x^2}=\frac1{a^2}$$ ...