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
vote
1answer
53 views

Real analysis homework problem

Let $h:[0,1] \times [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Assume that there is a constant $0<c<1$ such that $|h(x,y,s)-h(s,y,t)| \leq c|s-t|$ for all $x,y \in [0,1]$ ...
1
vote
2answers
85 views

Basic B-Spline basis function question

I am studying the basic recursion formula for generating B-Spline basis functions N(i,j) of a given degree from the basis for the lower degree, and puzzling at the magic. In particular what I am ...
1
vote
1answer
24 views

Find out alpha such that f is continous

Find out alpha such that f is continous in point 1: $$ f\colon\mathbb R\to\mathbb R, f(x) = \begin{cases} \frac{\ln(1 + \ln(2-x))}{(x-1)^\alpha},x \not= 1 \\ -1, x = 1. \end{cases} $$ ...
1
vote
2answers
55 views

How to prove $\frac{x^3}{x^2+y^2}$ is continuous?

$$ f(x,y) = \begin{cases} \frac{x^3}{x^2+y^2} & \text{ for } (x,y) \ne (0,0)\\ 0 & \text{ for } (x,y) = (0,0) \end{cases}$$ I know how to prove the function is ...
2
votes
1answer
71 views

What exactly is a modulus of continuity?

This is my first post on here, so forgive me if I am ignorant of certain customs. I am currently reading Courant's Introduction to Calculus and Analysis Volume I. Unfortunately, I have stumbled upon ...
0
votes
2answers
149 views

Prove that the Rational function $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ is uniformly continuous

I need some help with a calculus homework question. Here is said question: Let there be two polynomials $q$ and $p$ such that $\deg(p)\leq\deg(q)+1$ and $q(x)\neq0$ for all $x\in\mathbb{R}$. Show ...
0
votes
1answer
45 views

Follow-up regarding right-continuous $f:\mathbb{R} \to\mathbb{R}$ is Borel measurable

I have a follow-up to another question here on math.stackexchange, Are right continuous functions measurable?. The thread was a couple of years old, so I hope it's okay if I start a new question. ...
1
vote
2answers
310 views

Proving a complex function is continuous.

I've recently started complex analysis but I have very little background in complex numbers and to make sure I don't fall behind I'm doing some extra exercises one of which is Show $f$ is continuous ...
1
vote
2answers
71 views

Homework problem on continuity

Let $U =\{A \in M_{n} : A \text{ is invertible}\}$ (where $M_{n}$ is the space of all $n\times n$ matrices). $U$ is an open subset of $M_{n}$. Define $\alpha: U \rightarrow M_{n}$ by ...
1
vote
1answer
39 views

Why is the partial derivative $f_x' = 0 $ is not continous?

Looking again at my first CalculusII exam and I get confused about something. Let $ f(x, y) = \begin{cases} (x^2 + y^2) \sin\left(\frac{1}{x^2 + y^2}\right), & \text{if $(x, y) \ne (0, 0)$} \\ ...
1
vote
2answers
52 views

Uniform continuity of square root

I need to prove that $f(x)=\sqrt x$ is uniformly continuous on $[0, \infty)$. I wrote $\displaystyle |\sqrt{x}-\sqrt{c}|=|\frac{(\sqrt{x}-\sqrt{c})(\sqrt{x}+\sqrt{c})}{\sqrt{x}+\sqrt{c}}| \leq| ...
2
votes
1answer
50 views

Are all constrtuctively describable functions continuous? Do they necessarily come with a topology?

In the paper "An injection from $\mathbb{N}^\mathbb{N}$ to $\mathbb{N}$" by @AndrejBauer, about the question whether there exists an injection $\mathbb{N}^\mathbb{N}\to\mathbb{N}$, we writes ...
2
votes
1answer
45 views

Bilinear map on the set of finite sequences

Let $X = \{ x = (x_n)_{n=1} ^ {\infty}\subset \mathbb{R} \ \ | \ \ \exists N \in \mathbb{N} : \forall n>N : x_n=0 \}$ Let the norm on $X$ be $||x|| = \sum _{n=1} ^{\infty} |x_n|$ (which is fine, ...
0
votes
1answer
22 views

Continuous and binary variable question

For $y_1$ and $y_2$ as continuous variables how can this statement be reformed in binary and continuous variables with linear constraints Either $|y_1 - y_2| = 2$ or $|y_1 - y_2| = 4$
4
votes
1answer
57 views

Can any piecewise function be represented as a traditional equation?

In "Fundamentals of Electrical Engineering" we learned about piecewise functions for the "unit-step" and "ramp" which are represented by $f(x)= \begin{cases}0, & \text{if }x< 0 \\ 1, & ...
3
votes
2answers
99 views

Why is pointwise continuity not useful in a general topological space?

On page 27 of Lee's Introduction to Topological Manifolds, he writes In metric spaces, one usually first defines what it means to be continuous at a point...in topological spaces, continuity at a ...
-1
votes
3answers
113 views

product of two non zero continuous function is zero

Can you give me examples of two functions $f$ and $g$ such that both are non-zero continuous function but their product is zero.
2
votes
1answer
30 views

Why not define the Conway base-5 function, instead of base-13?

Evidently, the weird number 13 turning up in the definition of this function is just so there's 3 extra digits, in addition to the 10 decimal ones. But 10 itself sure is pretty arbitrary here, and ...
3
votes
3answers
437 views

Inverse image of a compact set is compact

Let $X$ and $Y$ be topological spaces, $X$ compact, $f : X \to Y$ continuous. Then the preimage of each compact subset of $Y$ is compact. With the stipulation that $X$ and $Y$ are metric spaces, this ...
0
votes
0answers
54 views

Composition of uniform convergent function is not uniform convergent

I am trying to come up with an example for the following situation: Say we have 2 sequences of functions $f_n:U \rightarrow R $ and $g_n: \rightarrow W $ both uniformly convergent to $f$ resp. $g$ ...
0
votes
1answer
40 views

Continuity of a piecewise function

Where is the function $$f: \mathbb R\to \mathbb R,\quad f(x)=\begin{cases}x^2 & x\le 0 \\x+1 & x\gt0\end{cases}$$ continuous? Question from my real analysis class. I know that it is ...
5
votes
4answers
88 views

Continuous function on $\mathbb{Q}$

Let $f:\mathbb{Q}\to\mathbb{R}$ be a function defined as: $$f(x) = \begin{cases} 0 & x^2 < 2\\ 1 & x^2 \geq 2 \end{cases} $$ Is this function continuous? How can we check the ...
5
votes
3answers
97 views

If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?

A problem that I have been thinking: For what kind of functions $f:[a,b]\to \mathbb R$ is $x\mapsto\sup_{t\in [a,x]}f(t)$ continuous? This is not true for bounded functions, take for example $f(x)=0$ ...
0
votes
1answer
42 views

limit of continuous function in complex plane

When I have a function, say $v$, continuous at $z_0$, then $\lim_{z\to z_0}v(z)=v(z_0)$. Does that imply that $\lim_{z\to z_0}iv(z)=iv(z_0)$, where $i$ is the imaginary number. Also, if I have u as ...
0
votes
1answer
68 views

Finding limit using polar/non-polar coordinates

Use polar coordinates to find the limit. [If (r, θ) are polar coordinates of the point (x, y) with r ≥ 0, note that r → 0+ as (x, y) → (0, 0).] (If an answer does not exist, enter DNE.) ...
1
vote
2answers
49 views

Integers seen as a continuous shape

In a popular maths book I find this sentence, in the context of an explanation of the difference between discrete and continuous, especially as regards groups: The group of integers is discrete; ...
0
votes
1answer
47 views

limit of continuous functions

If $f,g:[a,\infty)\to \Bbb R$ are continuous functions and $f$ is uniformly continuous on $[a,\infty)$ and $$\lim_{x\to \infty} (f(x)-g(x)) = 0,$$ how can I show that $g$ is uniformly continuous on ...
3
votes
0answers
49 views

Sufficient conditions for existence of injection from a metric space $M$ to $\mathbb{R}$

Let $M$ be any metric space. What conditions are required of $M$ for there to exist an injective, continuous function $$\varphi \colon M \longrightarrow \mathbb{R}$$ I would like to believe that ...
4
votes
1answer
486 views

If $f$ is twice differentiable and $f(2^{-n}) = 0 $, for all $n \in \mathbb N$, then $f^\prime(0) = f^{\prime\prime}(0) = 0$.

Let $f : \mathbb R \to \mathbb R$ be a twice differentiable function, such that $f(2^{-n}) = 0$, for all $n \in \mathbb N$ . Show that $$f^\prime(0) = f^{\prime\prime}(0) = 0.$$ My attempt. First, ...
3
votes
1answer
273 views

Continuity of an inverse function.

Theorem. Let $f\colon I \to J$, where $I$ is an interval and $J$ is the image $f(I)$, be a function such that: $f$ is strictly increasing on $I$. $f$ is continuous on $I$. Then $J$ is an interval, ...
0
votes
2answers
58 views

Proving a property of piecewise continuous functions

How to prove the following problem: Suppose $f \in PC(a,b)$, where $PC(a,b)$ means the set of piecewise continuous functions on the interval $[a,b]$ and $f(x) = \frac{1}{2}[f(x-) +f(x+)]$ for all $x ...
1
vote
3answers
34 views

How to show if a function is continuous

I am faced with this function: $$ h(x)= \frac{1−2x}{x^2−x−6} $$ And I have to determine its points of discontinuity. I am not quite sure how to approach this. Any help? Thank you!
1
vote
1answer
32 views

Continuity: Topology by Munkres

I can't seem to convince my self of this equality, assume that $f|U_{\alpha}$ is continuous for each $\alpha$, then if V is open in Y for Y being a topological space then, $f^{-1}(V) \cap ...
0
votes
1answer
29 views

Extension by continuity of $\frac{x-n \pi}{\sin(x)}$

I read that the function $$ \frac{x-n \pi}{\sin(x)} $$ is of class $C^{\infty}$ on a neighborhood of $n \pi$. What is meant ? Is that true ? It seems to me like this function can't be continuously ...
2
votes
0answers
101 views

Counterexample to Converse of Extreme Value Theorem?

The extreme value theorem says: If $X$ is a compact topological space, then for all functions $f: X \to \mathbb{R}$ such that $f$ is continuous we have that $f$ satisfies the extreme value property. ...
1
vote
2answers
54 views

How can I prove continuity of this function?

I have to prove that the function $f:\Bbb{R}\times\Bbb{R}\to\Bbb{R}$ defined as follows: $$f(x, y)=\frac{xy}{x^2+y^2}\text{for } (x, y)\neq (0,0)$$ $$f(x, y)=0\text{ for }(x, y)=(0,0)$$when taken as a ...
0
votes
1answer
51 views

How do I show $f(x)=\sqrt[3]{x}$ is continuous without compactness?

I'm a little lost on where to go after the using the identity $|\sqrt[3]{x_1}-\sqrt[3]{x_2}|=\frac{x_1-x_2}{|\sqrt[3]{x_1^2}+\sqrt[3]{x_1x_2}+\sqrt[3]{x_2^2}|}$.
0
votes
0answers
52 views

calculus continuity question.please help.

Show that $\sin (x + y)$ and $\cos(x-y)$ are continuous at $(0,\pi/2)$ using $\epsilon$ and $\delta$ definition. I have tried it as follows. Let $\epsilon>0$ be any real number. To find $\delta ...
1
vote
2answers
104 views

Questions about continuous functions.

Recently when working with my thesis, I've got 2 questions. Let $S_n$ be the set $\{x=(x_1,x_2,\cdots,x_n)\in\mathbb{R}^n\mid x_1+x_2+\cdots+x_n=1~\mbox{and}~0\leq x_i~\mbox{for}~ i=1,2,\cdots,n\}$. ...
1
vote
0answers
26 views

Continuity and Uniqueness Explanation?

I'm reviewing for an exam and I'm attempting to understand this problem (not homework): Verify that both $y_1(t) = 1 − t$ and $y_2(t) = −t^2/4$ are solutions of the initial value problem $$y' = ...
1
vote
1answer
34 views

Is $f(x) = x^{2}$ continuous on these topologies?

Just a quick question: $X = Y = \mathbb{R}$, $T_{X} = \{\emptyset , ]-\infty , 0[, [0, +\infty[, X\}$ and $T_{Y} = \{\emptyset , [0, +\infty[, X\}$. Does $f(x) = x^{2}$ define a continuous mapping? ...
0
votes
3answers
75 views

Calculus: Limit and continuity

Would anyone mind telling me how to solve these two questions? I know it sounds silly but I really have no idea.
2
votes
1answer
97 views

epsilon/delta definition alternative?

The phrase "any epsilon greater than zero" has always seemed somewhat vague. Question: is this an equivalent definition? $\forall\mbox{ Natural Numbers }N>0 \,\,\exists\delta>0\mbox{ s.t. ...
18
votes
1answer
264 views

$f(f(\sqrt{2}))=\sqrt{2}$ then f has a fixed point

$f(x)$ is continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ $f(f(\sqrt{2}))=\sqrt{2}$ Prove that $f$ has a fixed point in other words prove the there is $x_1$ such that $f(x_1)=x_1$ I tried using ...
0
votes
0answers
57 views

$f \in \mathcal{R}(\alpha)$ on $[a,b]$, then $\exists P_n$ s.t. $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$.

Assume $f \in \mathcal{R}(\alpha)$ on $[a,b]$, and prove that there are polynomial $P_n$ such that $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$. This is what I have, ...
1
vote
2answers
52 views

Continuity from complete metric space

Let $f:X\rightarrow Y$ be a continuous function, such as: $f(X)=Y$. If $X$ is complete, does it imply $Y$ is complete?
1
vote
1answer
51 views

Differentiability with non continuous partials (origin)

The function $$f(x,y)= \frac{x^{2}y^{2}}{(x^{2}+y^{4})} \quad if \quad (x,y) \neq (0,0)$$ $$f(0,0)=0$$ In order to study it's differenciability at the origin, I've studied if the partial are ...
3
votes
5answers
483 views

$\sqrt{x}$ isn't Lipschitz function

A function f such that $$ |f(x)-f(y)| \leq C|x-y| $$ for all $x$ and $y$, where $C$ is a constant independent of $x$ and $y$, is called a Lipschitz function show that $f(x)=\sqrt{x}\hspace{3mm} ...
1
vote
0answers
52 views

Continuity and differentiability of $x^a\sin ({1\over x}) $ at $0$

Consider the function $$ g_a (x) = \begin{cases} x^a\sin ({1\over x}) & x \neq 0 \\ 0 & x=0 \end{cases}$$ I am looking to determine for which $a$ the map $g_a$ is differentiable on ...
3
votes
1answer
155 views

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...