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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Show f$ \rightarrow D^\alpha f$ is a continuous mapping of $C^{\infty}(\Omega)$ into $C^{\infty}(\Omega)$

The question is from Rudin's Functional Analysis chapter 1 number 17. It is stated as follows. Show f$ \rightarrow D^\alpha f$ is a continuous mapping of $C^{\infty}(\Omega)$ into ...
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Oscillation Index of Thomae function

In this paper, for a real-valued function $f$ and a fixed $\epsilon>0$, the authors defined $$D(f,\epsilon,P)=\{ x \in P: \text{for all neighbourhood } N_x, \text{there exists } x_1, x_2 \in P ...
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Proving continuity of $f(x)=x\cos(2\pi/x)$ at $x=0$

I know that the function $f(x)=x\cos(2\pi/x)$ if $x\neq0$ and $f(0)=0$ is continuous at $x=0$ using $\epsilon-\delta$ as follows: $\lvert x\cos(2\pi/x)\rvert=\lvert ...
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Proving that a function $g:\Bbb{R}^2\rightarrow \Bbb{R}, (x_1, x_2)\mapsto g(x)=\frac{x_1x_2}{x_1^2+x_2^4}$ is not continuous at $x=0$

I would like to prove that the following function $g:\Bbb{R}^2\rightarrow \Bbb{R}$ is not continuous at $x=0$ $$ g(x)=\frac{x_1x_2}{x_1^2+x_2^4} $$ if $x\ne 0$ and equal to $0$ if $x=0$. It's pretty ...
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How to prove that a continuous mapping from a compact, connected space..

If $ f $ is a continuous mapping from a compact, connected metric space M to the real numbers and there exists a real number s such that f(m) never equals s, then there exists a constant $ c>0 $ ...
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44 views

Continuous function from a connected set?

If $ f $ is a continuous mapping from a connected set to the real numbers and there exists a real number s that nothing maps to, then the image is either greater than or less than s. this is clear to ...
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28 views

Continuty of functions inside a open ball

Let $ f: X \subset \mathbb{R}^p \to \mathbb{R}^q $ and $ a \in X$. Supose that for all $ \epsilon > 0 $ exists $ g: X \to \mathbb{R}^q $ continuous at $a$ such as $ \| f(x) - g(x) \| < \epsilon ...
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61 views

Show that an inverse mapping $ g = f^{-1}: J \to I $ is continuous

Question: Let $f: I \to \mathbb R $ be a continuous, injective funtion. Show that its inverse mapping $ g = f^{-1}: J \to I $ is continuous, where, $I = (x,y)$ and $J = f(I)$. I understand this ...
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$f$ is continuous and open implies $f$ injective

Question: Let $f: \mathbb R \to \mathbb R$ be continuous and open, that is if $A \subset \mathbb R$ is open then $f(A) \subset \mathbb R$ is open. Prove that $f$ is injective. Attempt: Suppose $f$ ...
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How can I prove that a continuous injective function is increasing/decreasing?

If I have a continuous, injective function mapping the real numbers, then it is either increasing or decreasing. This seems intuitively obvious but I can't come up with a neat proof for it.
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98 views

Prove that $f(z)=z^2$ is continuous.

Prove that $f(z)=z^2$ is continuous for all complex and real values of $z$. What I've got so far is: Given $ \epsilon >0$ and $|z-z_0|<\delta$ after some calculations (which I've checked with ...
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32 views

Mean value theorem sufficient conditions in several variables

I was doing a proof which was: If f is defined on an open set A, and all of its partial derivatives EXIST and are BOUNDED at A, then f is continuous. I used the trick of writng down (just to ...
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30 views

PDF of product of two continous joint distribution

Suppose that $X1$ and $X2$ have a continuous joint distribution for which the joint PDF is as follows: \begin{equation*} f(x_1,x_2) = \begin{cases} x_1 + x_2 & \text {for $0 < x_1 < ...
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1answer
23 views

Function continuity outside a closed subset

Let $f:M \subset \mathbb{R}^p \to \mathbb{R}^q $,continuous at $a \in M $. Show that if $f(a) \notin \overline{B} (b,r) \subset \mathbb{R}^q $, then exists $ \delta > 0 $ such as $ f(x) \notin ...
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128 views

Equality on functions in $ \mathbb{R}^n $

Let $ f,g : M \subset \mathbb{R}^p \to \mathbb{R}^q $ continuous. Given $ a \in M $, supose that all open ball centered in $a$ contains a point $x$ such as $f(x) = g(x) $. Show that $ f(a) = g(a) $. ...
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29 views

Continuity of the multiplication map $f\mapsto x^2 f(x)$ between normed spaces

Let $F:C[0,2]\to C[0,2]$ be the map defined by $(F(f))(x)=x^2f(x)$. Show that $F$ is continuous as a function from $(C[0,2],\|\cdot\|_{\sup})$ to $(C[0,2],\|\cdot\|_{2})$. I read this solution: ...
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81 views

Why upperbound $|x-a|$ by 1 in the proof of continuity?

In most (all?) proofs of continuity of polynomials ($x^2, x^3$, etc), for example in Max Rosenlicht's book (http://www.math.pitt.edu/~frank/pittanal2121.pdf, page 97), the usual trick is to get to the ...
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60 views

Preservation of inequality on continuous functions

Let $ f,g:M \subset \mathbb{R}^{p} \to \mathbb{R} $ countinuous function at $a \in M$. Show that if $f(a) < g(a)$ then exists $ \delta >0 $ such as for $x$ and $y$ in $M \cap B(a, \delta) $ ...
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69 views

Proving that a polynomial has a positive root

So I want to prove that a polynomial $ P(x)=a_nx^n+a_{n−1}x^{n−1}+.....+a_1x+a_0 $ has a positive root. I'm given that $ a_n $ is positive and $ a_0 $ is negative. I want to know how to apply the ...
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51 views

Splitting polygon in half. [closed]

Let $P$ be a convex polygon in the plane. Prove that there is a vertical line which splits P onto two polygons of equal area. I tried to use intermediate value theorem with no luck.
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show that the function below is linear [duplicate]

let $f$ be a continuous function from $R$ into $R$ with this property: $f(x+y) = f(x) + f(y)$, for all $x,y \in R$. Prove that $f$ is linear.
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Linear homotopy

Let $\lambda, \mu:[a,b]\longrightarrow X\subset\mathbb{R}^n$ paths such that the straight line $[\lambda(s),\mu(s)]$ lies in X for all $s\in[a,b]$. Set: $$\begin{array}{lccc} ...
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35 views

Continuous complex function from Rudin's Real and complex analysis

Lemma 10.29 from Rudin's Real and Complex Analysis, p. 314 of the third edition states that "if $f \in H(\Omega)$, then $g:\Omega \times \Omega \to \mathbb{C}$ defined by \begin{equation} g(z, w) = ...
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34 views

Additive function and continuity at a point

Does continuity at a point and Additive function imply continuity at all other points in a normed linear space. Is there some result like there exist a in field such that f(x) = ax for all x in normed ...
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17 views

Basic definition of continuity [duplicate]

Ltf(c+h) = f(c)(h goes to 0) if and only if Ltf(x) = f(c)(x goes to c). I am able to prove this fact using sequential criterion of continuity. But sequential criterion is dependent on Axiom of ...
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1answer
72 views

Connectedness, continuous functions, and the intermediate value theorem

I want to prove that for a continuous function mapping a connected space to ℝ such that f(p) never equals s, it follows that f(p) < s for all p or f(p) > s for all s. So here's what I know so ...
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41 views

Finding $a$ and $b$ so that the function is continuous

$$f(x) = \begin{cases} \displaystyle\frac{x^2-4}{x-2}&\quad x<2\\[0.4em] ax^2-bx+3&\quad 2 \leq x <3\\[0.3em] 2x-a+b&\quad x \geq 3 \end{cases}$$ I can't make the right limit of ...
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1answer
30 views

Find a function that matches the following conditions.

Find a function that matches the following conditions. (a) $f(x)$ is continuous for all real numbers (b) $f(0)$ = 3 (c) For all real numbers $x$, $f(x) = f(x/2)$ This is from a past paper, and the ...
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1answer
43 views

multivariable limit problem

I have a confusion regarding this problem. Problem: $\displaystyle f(x,y)=\frac{\sin^2|x+2y|}{x^2+y^2}$ is continuous for all $(x,y)\neq (0,0)$. True or false? I think that the limit does not exist ...
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112 views

Epsilon Delta Proof?

I always have trouble with understanding the intuition/process of $\epsilon$-$\delta$ proofs. Could anyone assist me with understanding the solution to the following: Show that $f$ is continuous at ...
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15 views

A real valued function having IVP

Given $f:\mathbb R\rightarrow \mathbb R$ be a function which maps intervals to intervals. Suppose for each sequence $x_n\rightarrow x \exists M $ such that $|f(x)-f(x_n)|\leq ...
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If $f$ and $g$ are uniformly continuous on $\Bbb{R}$ then $f\circ g$ is uniformly continuous on $\Bbb{R}$

Prove or disprove: If $f$ and $g$ are uniformly continuous on $\Bbb{R}$ then $f\circ g$ is uniformly continuous on $\Bbb{R}$. I think there's something crooked in my attempt. I would like to know what ...
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56 views

IVT question involving polynomial with even degree

Let $M(x)$ be an even polynomial with a positive leading coefficient, with $a_{2n} > 0, n\ge1 $. Show that there exists a constant $a*\in \mathbb{R}$ such that $M(x)+a = 0$ has a real root if ...
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1answer
113 views

How to define Square Root

I'm trying to understand how to define the square root of a complex function "globally". Let's say we have some function from some set $X$ onto $\mathbb{C} - \{0\}$: $$ f:X\to\mathbb{C}-\{0\} $$ and ...
2
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1answer
68 views

Prove for some $z_0 \in C$ the function $f(z)=|z-z_0|$ is continuous on all of $\mathbb{C}$

Let $z_0\in\mathbb{C}$ and $f(z)=|z-z_0|$. Show that $f$ is continuous on $\mathbb{C}$. I expect to see a proof using the triangle inequality. Note a function $f$ is continuous on $\mathbb{C}$ if ...
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1answer
44 views

How to prove this statement?

I cannot prove this proposition directly . Let $(X,d)$ and $(Y,d')$ be metrice spaces. Let $f$ be a function from $X$ to $Y$. If $\overline{f^{-1} ( B)} \subseteq f^{-1}( \overline B)$ for all ...
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1answer
22 views

Questions on continuously differentiable function on $[a,b]$

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Normally we define derivatives of $f$ only at interior points in $[a,b]$. But when we write $f\in C^1([a,b])$, it means that $f$ is differentiable on ...
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101 views

Question about the application of continuous functions and IVT

I came across a question which says: Suppose that $f:[0,2 \pi] \to \mathbb{R}$ is continuous, and $f(0)=f(2 \pi)$. 1.Show that there exists $x \in [0,\pi]$ such that $f(x)=f(x+ \pi)$. ...
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Finding all continuous functions so that $f^n(x)=x$ for some $n$.

I came up with this problem in class but I can't seem to solve it. I need to find all the functions $f$ with domain and codomain $\mathbb R$ such that there is an $n$ such that $f^n(x)=x$ for all $x$, ...
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39 views

Find the points where the function is continuous

Let $X \subset \mathbb{R}$ be a finite set and define $f: \mathbb{R} \to \mathbb{R}$ by $$ f(x)= \begin{cases} 1 & \text{if $x\in X$},\\ 0 &\text{otherwise}. \end{cases} $$ At which points ...
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If a continuous function on $\mathbb{R}$ $f$ receives an extremum at a single point, it must be the global extremum.

Let $f$ be a continuous function on $\mathbb{R}$ which attains a local maximum at ${{x}_{0}}$. Prove that if $f$ doesn't have any other extremum points, then ${{x}_{0}}$ is the global maximum of $f$ ...
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1answer
41 views

Is this theorem about integration with substitution wrong?

A theorem in my book states: If $g$ is differentiable, f is continuous, and F is an antiderivative of f, then : $\int f[g(x)]g'(x)dx=F[g(x)]+C$ The reason I am asking if this is correct, ...
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1answer
38 views

Functions and continuity proof in real analysis

Prove: If $f\colon A\rightarrow\mathbb{R}^m$ and $a\in A$, show that $\lim_{x\rightarrow a}f(x)=b$ if and only if $\lim_{x\rightarrow a}f^i(x)=b^i$ for $i=1,\dots,m$. The end of the statement is ...
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66 views

Does Intermediate Value Theorem $\rightarrow $ continuous?

i try to understand Intermediate Value Theorem and wonder if the theorem works for the opposite side. I mean, if we know that $\forall c\:\:\:f\left(a\right)\le \:c\le \:f\left(b\right)\:,\:\exists ...
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33 views

Checking when an $a$-dependent function is continuous, differentiable.

For some $a\in \Bbb{R}$ define a function $f_{a}(x) = \begin{cases} {x^{a}\cos{1\over x}}, & \text{if $x$ $\ne$ 0} \\[2ex] 0, & \text{if $x=0$} \end{cases}$. Hints firstly are preferred. b. ...
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12 views

continuity with 2-variables

The question is Determine whether $f$ can be defined at $(0,0)$ so that is is continuous $$f(x,y) = \frac{x^py^q + x^ry^s}{x^qy^p + x^sy^r}, p,q,r,s > 0$$. I chose numbers for p,q,r,s and ...
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1answer
37 views

What variables is $\delta$ dependent on in the epsilon-delta definition of continuity?

The definition of continuity is: $f$ is continuous at $a$ if: Given any $\epsilon>0 $, $\exists \delta > 0$ st. $|x-a|<\delta \implies |f(x)-f(a)|< \epsilon$ $\delta$ obviously depends ...
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86 views

Which of these statements about a continuous function is true? [closed]

A function $f$ is continuous on the interval $[0, 2]$. It is known that $f(0) = f(2) = -1$ and $f(1) = 1$. Which one of the following statements must be true? (A) There exists a $y$ in the interval ...
2
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2answers
85 views

Example of continuous function over $\mathbb R^n$

Let $f:[0,1]\to\mathbb R^n$ such that $f(t)=ty+(1-t)x$ for some $x,y \in \mathbb R^n$. Prove that $f$ is continuous. I know a definition that A function $f\colon X \rightarrow Y$ between two ...