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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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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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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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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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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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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2answers
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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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35 views

Splitting polygon in half. [on hold]

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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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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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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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
37 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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1answer
34 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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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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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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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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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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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
80 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 ...
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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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38 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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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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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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2answers
42 views

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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35 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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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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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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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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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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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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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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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 ...
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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 ...
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1answer
30 views

How to tell if a function is continuous at (0,0)

I have to decide if the following function is continuous at (0,0). it's f(x,y) = x^2 + y^2 if (x,y) does not = 0, and f(x,y) = 2 if (x,y) = (0,0) so for the first one, I assume it is continuous, ...
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Continuous functions and open sets

I'm working on a proof and having trouble applying a certain theorem. I want to prove that if $ f $ is a continuous function from a metric space into the real numbers, then the set $ {f(x)>0} $ ...
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Proof: Cauchy sequences and uniform continuity

I'm working on a proof and I'm having trouble relating definitions I want to prove that if f is uniformly continuous, then if a sequence $ {a_n} $ is Cauchy, $ {f(a_n)} $ is Cauchy. So if $ f $ is ...
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29 views

Continuous and additive function is linear [duplicate]

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and $f(x+y)=f(x)+f(y)$, show that $f(x)=kx$, $k\in \mathbb{R}$. I tried to define $g(x)=f(x)-kx$ and $g(0)=0 $ but don't know how to ...
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Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
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652 views

Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty) $ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
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Riemann Integral Property for Continuous, Monotonic, Non-negative Function

If $f$ is continuous, non-negative, and monotonically increasing function on $[0,∞)$, then prove that $\int^{x}_{0} f(t)dt\leq xf(x)$ $\forall x ≥ 0$ My attempt: Define $F(x)=\int^{x}_{0} f(t)dt$. ...
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solve equation with Intermediate value theorem…

set $a_1$,$a_2$,$a_3>0$ and $λ_3>λ_2>λ_1$ on $ℝ$. show that there are exactly two $x$’s for $a_1/(x-λ_1) + a_2/(x-λ_2) + a_3/(x-λ_3) = 0$ I tried use the intermediate value theorem but I ...
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31 views

The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that f is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So ...
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31 views

three elementary problems on limits of several variable . [closed]

I'm learning limits of several variable new. Can anyone help me? Computing the following limits: $\lim_{(x,y)\to(0,0)}|x|^y$ $\lim_{(x,y)\to(0,0)}\sin(x/y)$ $\lim_{(x,y)\to(0,0)}x^2\cdot ...
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2answers
31 views

Properties for functions $f:[a,b] \to \mathbb R$? [closed]

Let $f:[a,b] \to \mathbb R$ be a function. Which of the followings are true: A) If $f(x)$ is continuous then it is bounded. B) If $f(x)$ is continuous then it is increasing. C) If ...
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35 views

Questions about absolutely continuous function

It could be a silly question but the definition of absolutely continuous function says that "A real valued function $f:[a,b]\rightarrow\mathbb{R}$ is absolutely continuous on $[a,b]$ if for all ...
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4answers
44 views

Prove a Continuous Distribution Function is Uniformly Continuous

Let $F$ be the distribution function for a random variable $X$ and it is given that $F$ is continuous over the entire real line. Prove that $F$ is uniformly continuous over the real line. My ...