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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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 ...
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37 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 ...
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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 ...
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13 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 ...
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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 ...
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19 views

Lipschitz continuity of continuously differentiable function

Is it true that a continuously differentiable function in a Banach space $X$ is locally lipschitz in $X$?
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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' ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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1answer
57 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 ...
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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 ...
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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 ...
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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 $f(y)=c\operatorname{sign} y$ ...
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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 ...
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62 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 ...
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12 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 ...
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1answer
66 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 ...
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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 ...
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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 $x_{0}\in[0,1]$ ...
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47 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 ...
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1answer
29 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 $h : X \to Y$ ...
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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?
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How to calculate Hyperoperators with reals? [on hold]

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?
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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 ...
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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 $f(x) = \{0: x<0, 1: x \geq 0 \}$, ...
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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 ...
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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 $\{0,1\}$. ...
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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 ...
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1answer
29 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 ...
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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 ...
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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 $\geq 1/2$. ...
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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 ...
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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 $f$ ...
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55 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 ...
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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 ...
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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 $x$. ...
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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: $$lim_{x\to a}\frac1{x^2}=\frac1{a^2}$$ ...
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5answers
23 views

Prove the continuity on an open interval

I need to show, that function $f(x) =\frac{2x +3}{x-2}$ is continuous on the interval $(2,\infty)$ My attempt: We should find the right-hand limit to prove the continuity: and this limit is equal to ...
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2answers
30 views

If f*g is Riemann integrable, g continuous, nonzero and bounded, show that f is Riemann integrable

How would I go about proving that if $fg$ is Riemann integrable, given that $g$ is continuous, nonzero, and bounded (so $g$ Riemann integrable), how would I go about showing that $f$ is Riemann ...
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36 views

Function is continuous if graph is compact.

Let $X$ be a Hausdorff space and let $f:X\to \mathbb{R}$. If grapph of $f$ is compact we have to show that $f$ is continuous. Since every closed subset of a Hausdorff space is closed, therefore ...
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60 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 $x_0 = a, a > 0$. To what I understand of this question, is it just asking to me ...
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17 views

let $f$ be a function defined on all of $\mathbb{R}$

Is there a function that is continuous in this specific manner? $\forall \epsilon >0$ we can choose $\delta = \epsilon$, and it follows that $|f(x) - f(c)| < \epsilon$ whenever $ |x-c| < ...
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33 views

Continuity, algebraic and rational numbers [closed]

Is it true that there exist a continuous function f that for every algebraic number q , his image f(q) is a rational number? Thank you for your answers
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12 views

Comparing smoothness among approximations

We are interpolating a missing fragment of a 2D curve given a set of sample points. Our method generates several candidates of curve pieces to fill the missing part, but we want to select the solution ...
2
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3answers
48 views

Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with "one-sided" limits or derivatives. This is something I find ...
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1answer
34 views

Prove $\{f_n(x)\}$ is not continous

$f_n(x) = \left\{ \begin{array}{ll} \frac{1}{n} & \quad x \in \mathbb{Q} \\ 0 & \quad x \notin \mathbb{Q} \end{array} \right.$ Not sure how to show ...