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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Intermediate value theorem (IVT) for a function

My teacher said that the function defined by: \begin{equation} f(x)=\begin{cases} \dfrac{1}{x}, & \text{if $x \neq 0$}.\\ 0, & \text{if $x = 0$}. \end{cases} \end{equation} is a ...
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1answer
54 views

Why is $f(x)=\frac{1}{x}$ discontinuous?

A continuous function $f:X\to Y$ is that which satisfies the following property: for open set $U\subset Y$, $f^{-1}(U)$ is open in $X$. We know that the function $f=\frac{1}{x}$ is discontinuous at ...
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1answer
27 views

Understanding why an IVP has a solution, using uniqueness and existence theory

Given the existence and uniqueness theory: If $f$ is Lipschitz continuous over some region $D$, then there is a unique solution to the initial value problem (IVP): $u'(t) = f(u,t), \hspace{5mm} ...
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2answers
32 views

Determine best possible Lipschitz constant

I'm slightly confused by a homework problem here...I've been given the function: $ f(u) = log(u) $ With the bounds: $ 2 \leq u \lt \infty $ Now I thought I understood what the Lipschitz Condition ...
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1answer
67 views

Continuous function, mapping of a set to itself [duplicate]

Let $f: [0,1] \rightarrow [0,1]$ be continuous. Any idea on how we can prove that it is not possible for $f$ to map $[0,1]$ onto $[0,1]$ exactly two-to-one. That is, there is no continuous $f$ as ...
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1answer
42 views

How can a graph have a jump discontinuty at $x \not= 0$ and removable discontinuity at $x = 0$?

I stumbled an a past test question where the student was asked to provide any example of a graph that has those two particular properties. I now there should be a piece-wise function, an maybe a case ...
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0answers
24 views

Uniform continuity of xsin(x)

Define $f:(0,\infty) \rightarrow \mathbb{R} $ by $f(x) = x^{\alpha} \sin(x^{\beta})$ for $\alpha, \beta > 0$. a) For what $\alpha, \beta$ is $f$ continuous? I've shown that $f'$ is bounded on ...
3
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1answer
181 views

Normed Vectors Spaces

Let $(E,\| \cdot \|_E)$ and $(F,\| \cdot \|_F)$ be two normed vector spaces over $\mathbb{C}$ and let $u: E\rightarrow F$ be a linear map. (a). Prove that the following conditions are equivalent: i. ...
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2answers
169 views

Help me correct my ideas of continuity

I've been studying real analysis over the past few months, and I'm having trouble organizing the different notions of continuity and ideas related to continuity in my head geometrically. I will ...
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1answer
44 views

Topology - interval homeomorphic to another interval

{a.} Prove that any open interval $(a, b)$ is homeomorphic to the interval $(0, 1)$. Define $f:(a, b) \to (0, 1)$ by $f(x)=(x-a)/(b-a)$, which is one-to-one and onto. Consider $f^{-1}:(0, 1) \to (a, ...
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1answer
37 views

Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$

$f(x)$ is continuous on $[a,b]$. Now we define a new function $M(t)$, for every $t\in[a,b]$ $$M(t) = \sup_{a \leq x \leq t} f(x).$$ Prove formally that $M(t)$ is continuous on $[a,b]$. (sup = ...
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2answers
36 views

Theorem: Let (X, t) and (Y, u) be topological spaces, let (A, tA) be a subspace of (X, t) and let (B, uB) be a subspace of (Y, u).

Theorem: Let $(X, t)$ and $(Y, u)$ be topological spaces, let $(A, t_A)$ be a subspace of $(X, t)$ and let $(B, u_B)$ be a subspace of $(Y, u)$. If $f:X\to B$ is continuous, then the function $g:X\to ...
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21 views

Discontinuity and open sets

I came across the following definition by reading the book "Mostly Surfaces": The map $f: X \to Y$ is continuous if it has the following property: For any open $V \subset Y$ the set $$U = ...
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1answer
25 views

Differentiability of trigonometric piecewise functions

So I have a function of a real variable $x$: $f(x) = \left\{\begin{array}{lr} x \int_0^{tanx} \dfrac{t^2}{\sqrt{1+t^3}}dt & if \: x \ge 0\\ sin^2(x) & if \: x \lt 0 ...
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1answer
28 views

The Group of Complex Continuous Functions?

Let $C(\mathbb{C},\mathbb{C})=\{f:\mathbb{C} \rightarrow \mathbb{C}\,|\,f $continuous $\}$ be the set of all continuous functions from the complex plane to itself and consider the composition ...
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1answer
63 views

Prove continuity for cubic root using epsilon-delta

I am trying to prove that a function is continuous at a point a using the $\epsilon$-$\delta$ theorem. I managed to find a $\delta$ in this case $|2x^2+1 - (2a^2+1)| < \epsilon$. But I have a hard ...
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2answers
45 views

How do i prove that $f(x,y)=y-x$ is continuous? [closed]

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}:(x,y)\mapsto y-x$ be a function. How do I prove that $f$ is continuous?
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19 views

Regularising a function that is constant on an interval (related to Heaviside)

Define the function $f:\mathbb R \to \mathbb R$ by $$f(x) = \begin{cases} x &\text{for $x < 0$}\\ 0 &\text{for $x \in [0,1]$}\\ x-1 &\text{for $x > 1$}& \end{cases} $$ Note that ...
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2answers
84 views

Showing $f(x)\le 2$ for all $x\in [0,1]$

Let $F:[0,1]\to \mathbb R$ be continous such that for all $x\in \mathbb Q \cap [0,1]$ we have $f(x)\le 2$. Show that for all $x\in [0,1]$ we have $f(x)\le 2$. I'm going to write the answer I got ...
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0answers
46 views

How to prove a function is continuous?

What is a general outline of a proof of the continuity of a function? My background is Calculus 2. I think it uses the $\epsilon$-$\delta$ definition of a limit, but I only have a very vague idea of ...
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39 views

Proof that order of integration does not matter for non-continuous functions

For continuous functions of several variable, by Young's theorem the order of integration (i.e. "antiderivation") does not matter so that for instance $$ \int \left(\int f(x,y)~ dx\right) ~dy = \int ...
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49 views

Arithmetic property of continuous functions [closed]

Let $X$ be a topological space and $f, g :X \to \mathbb{R}$ be two continuous functions on $X$. Are the functions $f+g$, $f \cdot g$ continuous?
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3answers
112 views

Does every continuous function have a left and right derivative?

I understand that differentiability implies continuity, whereas the converse isn't true. But must a continuous function have both a left and right derivative, not necessarily equal to one another?
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1answer
48 views

Can such a function be continuous? [duplicate]

Let $f$ be a function from $\Bbb R$ to $\Bbb R$ such that $f(x)$ is rational when $x$ is irrational, and $f(x)$ is irrational when $x$ is rational. Can $f$ be continuous? Thanks for your help.
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4answers
88 views

Is $f(z)=\bar{z}$ continuous?

I have $z\in \mathbb{C}$, is $f(z)=\bar{z}$ continuous on the whole complex plane? Note that $\bar{z}$ is the conjugate of $z\in \mathbb{C}$ I was thinking that if $z$ is on the real line, then ...
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4answers
83 views

Why is every such function constant?

Let $f:[0,1] \to \mathbb{R}$ be a continuous function such that $f(x)=f(x^2)$ for all $x \in [0,1]$. Any hint/idea for proving that $f$ has to be constant?
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4answers
52 views

Boundedness of continuous functions.

I am trying to prove: Suppose that $f: \Bbb{R} \to \Bbb{R}$ is continuous on $\Bbb{R}$ and that $$\lim_{x \to \infty} f(x) = 0$$ and $$\lim_{x \to -\infty} f(x) = 0$$ Prove that f is bounded ...
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1answer
53 views

Prove that a plane is continuous?

Okay, so the problem gives a matrix: $$ \pmatrix{ 1&2\\ 3&4} $$ and this matrix is an $\Bbb R^2 \to \Bbb R^2$ linear map. I am asked to explicitly write the component functions of $A$, and ...
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734 views

A game with $\delta$, $\epsilon$ and uniform continuity.

UPDATE: Bounty awarded, but it is still shady about what f) is. In Makarov's Selected Problems in Real Analysis there's this challenging problem: Describe the set of functions $f: \mathbb R ...
5
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1answer
130 views

Extended matrix function

I have a continuous matrix-valued function $f:\mathbb{R}^d\mapsto {\cal M}_{k\times d}$, with $d<k$, such that $f(x)$ is full rank for all $x\in\mathbb{R}^k$. Can I extend this function to be a ...
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2answers
31 views

Continuity of $f(z)=u(x,y)+iv(x,y)$

If $u(x, y)$ and $v(x, y)$ are continuous (respectively differentiable) does it follow that $f(z) =u(x, y) + iv(x, y)$ is continuous (resp. differentiable)? If not, provide a counterexample. This ...
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1answer
45 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 $x,y \in [0,1]$ ...
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2answers
42 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 ...
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1answer
23 views

Find out alpha such that f is continous

Find out alpha such that f is continous in point 1: $$ f\colon\mathbb R\to\mathbb R, f(x) = \begin{cases} \frac{\ln(1 + \ln(2-x))}{(x-1)^\alpha},x \not= 1 \\ -1, x = 1. \end{cases} $$ ...
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2answers
51 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 ...
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1answer
54 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 ...
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2answers
96 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 ...
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1answer
24 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. ...
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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 ...
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2answers
63 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 ...
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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 $(x, y) \ne (0, 0)$} \\ ...
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47 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| ...
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1answer
47 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 ...
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1answer
41 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, ...
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1answer
20 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 $|y_1 - y_2| = 4$
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1answer
46 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
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1answer
54 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 ...
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3answers
66 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
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1answer
27 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
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3answers
219 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 ...