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)

3
votes
1answer
17 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 ...
1
vote
1answer
38 views

How can I prove that the inverse mapping of a continuous, injective function is continuous?

If I have a continuous, injective function $ f $ that maps from $ (x,y) $ in the real numbers into the real numbers, then its inverse mapping $ f^{-1} $ is continuous. I understand this intuitively ...
5
votes
1answer
37 views

$f$ is continuous and open implies $f$ injective

Question: If $f: \mathbb R \to \mathbb R$ is 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 $f$ ...
3
votes
3answers
27 views

How can I prove that a continuous injective function is strictly monotonic?

If I have a continuous, injective function from an interval [x,y] in the real numbers to the real numbers, then it is strictly monotonic. This seems intuitively obvious but I can't come up with a ...
3
votes
1answer
64 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 ...
1
vote
0answers
12 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 ...
0
votes
0answers
14 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 < ...
1
vote
1answer
21 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 ...
1
vote
1answer
35 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) $. ...
0
votes
2answers
24 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: ...
2
votes
2answers
40 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 ...
2
votes
1answer
24 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) $ ...
2
votes
2answers
50 views

Proving that a polynomial has a positive root

So I want to prove that a polynomial $ P(x)=a_nx^n+a_{nāˆ’1}x^{nāˆ’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 ...
1
vote
1answer
38 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.
0
votes
0answers
27 views

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.
1
vote
1answer
8 views

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} ...
0
votes
0answers
20 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) = ...
0
votes
1answer
23 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 ...
1
vote
0answers
16 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 ...
1
vote
1answer
38 views

Connectedness, continuous functions, and the intermediate value theorem

I want to prove that for a continuous function mapping a connected space to ā„ 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 ...
0
votes
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 ...
1
vote
1answer
24 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 ...
1
vote
1answer
23 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 ...
3
votes
3answers
69 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 ...
0
votes
0answers
14 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 ...
3
votes
1answer
25 views

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 ...
1
vote
2answers
40 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 ...
1
vote
1answer
95 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
votes
1answer
43 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 ...
0
votes
1answer
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 ...
1
vote
1answer
16 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 ...
1
vote
0answers
68 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 $f(x)=f(x+ \pi)$. ...
0
votes
2answers
43 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 $x$, ...
0
votes
2answers
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 ...
0
votes
2answers
40 views

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 $f$ ...
1
vote
1answer
37 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, ...
0
votes
1answer
33 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 ...
0
votes
1answer
113 views
2
votes
2answers
58 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 ...
0
votes
2answers
29 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. ...
0
votes
1answer
11 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 ...
3
votes
1answer
28 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 ...
-2
votes
1answer
39 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 ...
1
vote
2answers
42 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 ...
0
votes
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, ...
0
votes
2answers
26 views

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} $ ...
1
vote
1answer
32 views

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 ...
0
votes
0answers
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 ...
5
votes
1answer
32 views

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 ...
13
votes
1answer
654 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 $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$