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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Are the two statements about continuous functions equivalent?

I have always wondered about this: A continuous function is defined thus: for any $\epsilon>0$, there exists $\delta\in\Bbb{R}$ such that $|x-y|<\delta\implies |f(x)-f(y)|<\epsilon$ for ...
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34 views

$\varepsilon$-$\delta$ proof of continuity of floor function $\lfloor x\rfloor$

I would just like to ask someone to confirm or correct the following 'proof' of continuity of the floor function. Let $\varepsilon>0$ be given. Set $\delta:=\min\lbrace x-\lfloor x\rfloor,\lceil ...
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1answer
14 views

continuity of a function f = (f_1,f_2) in a product topology if f_1 and f_2 are continous

Say $X$, $Y_1$ and $Y_2$ are topological spaces. Let $f_1 \; X \to Y_1$ and $f_2 \; X \to Y_2$. If $f\; X \to Y_1 \times Y_2 $ $f(x) = (f_1(x), f_2(x))$ $Y_1 \times Y_2$ is a topological space with ...
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Find $\alpha$ and $\beta$ so that $f(x)$ is continuously differentiable

The function $f(x)$ is defined as following $$ f(x) := \begin{cases} \cos x+e^x, & \text{if $x < 0$} \\ \ \alpha(1+x)^{2009}+\beta e^{-x}, & \text{if $x \ge 0$} \end{cases} $$ I need to ...
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31 views

Explaining the one-dimensional continuity equation with respect to density evolution

I've got a rather abstract question So the continuity equation for a one-dimensional continuum is: $$ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}(\rho v)=0 $$ and we can expand ...
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25 views

Continuous partial derivatives

I have the following function and I want to show that it is differentiable. I am going to do this by showing that the partial derivatives are continuous and so I will show that they are continuous at ...
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16 views

Technicality: convergent sequence is preserved by continuous map

I'm looking at the proof of a basic result in Maxwell Rosenlicht's analysis book: Let $(E,d)$, $(E',d')$ be metric spaces. Let $f: E \to E'$ be a function. Then if, for every sequence of points ...
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32 views

Combination of continuous and discontinuous functions

I know that combining two continuous functions gives a continuous function, i.e., if $f(x)$ and $g(x)$ are continuous, then $f(x)\pm g(x)$, $f(x)\times g(x)$ and $f(x)\div g(x)$ are continuous ...
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1answer
43 views

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarrily lipschitz.

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarily lipschitz. Is the above statement true? I thought since $f$ is continuous on a compact metric space, $f$ ...
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2answers
69 views

Why can a discontinuous function not be differentiable?

I don't really understand why a discontinuous function cannot be differentiable. In Stewart's Calculus, the definition of a function $f$ being differentiable at $a$ is that $f'(a)$ exists. Earlier it ...
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58 views

Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
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The map that sends $A$ to its greatest eigenvalue is continuous.

The map $f:S_n(\mathbb R)\to \mathbb R$ such that $f(M)$ is the greatest eigenvalue of $M$ is continuous ($S_n(\mathbb R)$ is the set of symmetric matrices) I need to prove this result in order ...
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1answer
55 views

Modifications of Weierstrass's continuous, nowhere differentiable functions

Recalling how nowhere continuous functions such as the Dirichlet function can sometimes be modified on a $\lambda$-null set of points (in this instance, a countable set) to become everywhere ...
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18 views

Questions regarding regulated functions

I have a few quick questions regarding regulated functions: Firstly, I'll state the definition I have been given: Definition: Let $I=[a,b]$ be a compact interval. Then $f:I\to \mathbb{R}$ is called ...
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42 views

showing that $f(x,y)$ is continuous at $(0,0)$

Let $$f(x,y) = \begin{cases} 0, & \text{if $y \le 0$, $y \ge x^2$ } \\[2ex] 1, & \text{if $0 \lt y \lt x^2$ } \\ \end{cases}$$ Show that $f(x,y) \to 0$ as $(x,y) \to (0,0)$ along any ...
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Functions that are continuous only at two points?

I need to find a function $f:\mathbb{R}\to\mathbb{R}$ which is continuous only at two points, but discontinuous everywhere else. How on earth would I go about doing this? I can't think of any ...
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68 views

Prove $f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ defines a continuous function on $\mathbb{R}$.

Prove $$f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k$$ defines a continuous function on $\mathbb{R}$. I think we can show that if $\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ is uniformly ...
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Proof verification: $\int_a^x f(t) \text{dt}=0$, $f$ is continuous at $x$. Prove that $f(x)=0$

Let $f:[a,b]\to R$ be an integrable function such that for all $x \in[a,b]$, we have $\int_a^x f(t) \text{dt}=0$. Show that if $f$ is continuous at $x \in [a,b]$, then $f(x)=0$. My attempt: argue ...
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Show that this function is continuous at $(0,0)$

In this case i'm struggling to show that the partial derivatives with respect to x are continuous. The answers always brush over how you determine it like it trivial so i think i'm missing something. ...
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2answers
64 views

Is $T':L^2(\Omega) \to L^2(\Omega)$ continuous?

Here, $k$ is a fixed number. Let $$T(x) = \begin{cases} -k &x \in (-\infty, -k]\\ x &x \in (-k, k)\\ k &x \in [k, \infty) \end{cases}.$$ So $$T'(x) = \begin{cases} 0 &x \in (-\infty, ...
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79 views

If $f'$ exists where f is a function from $(0, \infty)$ to $R$, then $f$ is continuous on $(0, \infty)$? [closed]

If $f'$ exists where f is a function from $(0, \infty)$ to $R$, then $f$ is continuous on $(0, \infty)$? If it is true, then how can you prove it with epsilon-delta definition??
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Where rational functions are undefined

I have another question/comment I'd like a fresh pair of eyes on The question is "A rational function can have infinitely many x-values at which it is not continuous" I know since Q(x) in the ...
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44 views

Construct a continuous function which has no derivative almost everywhere.

Georg Cantor is famous for the first set theory (in "naive" terms) and the diagonal argument. However Cantor is also credited with the Cantor Set and for constructing a continuous function which has ...
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23 views

help with continuous and differentiable theorems

consider $b>a>0$ and $f:\left[a,b\right]\rightarrow \mathbb{R}$. $f$ is continuous at $\left[a,b\right]$ and differentiable at $\left(a,b\right)$. also, ...
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About continuity

one question is disturbing me : let f and g two continuous (real-valued) functions on the unit interval [0,1] with the property that $[f(x)-f(y)][g(x)-g(y)]=0 ,\forall x,y \in [0,1]$. To my ...
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Finding the continuity of the mapping of a solution to a PDE to its partial derivative

Here is a modified version of the Black-Scholes PDE: $\frac{\partial \phi(t,S,i)}{\partial t}$ + $r_iS\frac{\partial \phi(t,S,i)}{\partial S}$ + $\frac{1}{2} \sigma^2_i S^2 \frac{\partial^2 ...
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Determination of $ C^0 , C^1 , G^0 \text{and } G^1 $ Continuity?

Given two curves $ r(t) = (t^2-2t+2 , t^3,2t^2+t ) $ and $ n(t) = (t^2+1,t^3) $ where $ 0<=t<=1 $ . How can I determine $ C^0 , C^1 , G^0 \text{and } G^1 $ Continuity ?
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On continuous functions and second derivative

Let $f:[a,b]\to\mathbb R$ be a continuous function suh that $f''(x)$ exists $\forall x\in(a,b)$ . If $a<c<b$ and $f(a)=f(b)=0$ , then how to show that $\exists d\in(a,b)$ such that $f(c)=\dfrac ...
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1answer
34 views

Prove that any monotonic and bijective function is an homeomorphism with the usual topology

Let $f:(\mathbb{R},d_u\longrightarrow(\mathbb{R},d_u)$ an arbitrary function. Where $d_u$ is the usual distance. I have an exercise in which, with the sole asumptions of it being monotonic ...
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34 views

Proving that an arbitrary function is continuous (via topological methods)

Consider the following applications: $$f:(X,d)\longrightarrow(\mathbb{R^2},d_u)$$ $$\pi_i:(\mathbb{R^2},d_u)\longrightarrow(\mathbb{R},d_u)$$ Where $f$ is an arbitrary function, $\pi_i$ is the ...
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1answer
55 views

Is continuous extension on dense subset an isometry

If we have that $X \subset V$ is dense linear subspace. Where $V$ is normed space. I can show that for any $f \in X^{*}$, there exists a unique extension $\bar{f}$. I want to know if it can be shown ...
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Investigating a function with a parameter

I got stuck on solving this problem: For which $a \in \Bbb R$ is the function $$ f_a: \ ]1, \ \infty[ \; \longrightarrow \ \Bbb R: x\mapsto \frac{\log x}{(x-1)^a} $$ continuous on $[1, \ ...
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112 views

Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
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113 views

$f:\mathbb R \to \mathbb R$ is continuous and lim$_{n\to \infty} f(nx)=0$ for all real $x$ $\implies $ lim$_{x \to \infty}f(x)=0$

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that for all real $x$ , lim$_{n\to \infty} f(nx)=0$ , then how to prove that lim$_{x \to \infty}f(x)=0$ ? Please help and please don't ...
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1answer
19 views

Define multiple-variable function to be continuous

Define the function $f(x,y)= {{x^2 + y (x^2 + y)} \over {x^2 + y^2}}$ at $[0,0]$ so that the function would be continuous. I need help with this calculus problem. I mean, I guess it involves some ...
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68 views

Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$

Let $f\in C^{\infty} ([-1,1])$ and suppose there's a constant $M>0$ such that $|f^{(j)}(x)|\le M:\forall j\in\mathbb N$ (including zero) and for all $x\in [-1,1]$. Prove that if $f(\frac 1 ...
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26 views

Continuity and differentiability of a function. [duplicate]

How to prove that $f(x)$ is discontinuous at $x=0$ ? $$f(x)=\begin{cases} \sin\left(\tfrac1x\right),& \text{when $x\neq0$} \\ 0, & \text{when $x=0$} \end{cases}.$$
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Show for whih values this following function is continuous

For the function $f: [0,2 \pi] \rightarrow \mathbb{R}$ ,state at which points $c \in [0, \pi]$ is $f$ continuous or discontinuous. $$f(x)=\begin{array}{cc} ( & \begin{array}{cc} ...
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206 views

Proof that if f is function, continuous on an interval I then f(I) is also an interval

The theorem would be: Let $f:E\to\mathbb{R}$ a continuous function and $I$ and interval, $ I \subseteq E $. Then $f(I)$ is also an interval. I'm not sure if I've understood completely what I have to ...
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34 views

Continuity proof of two-variable function.

The Assignment Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} ...
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Determine if the following function is continuous in $(0,0)$.

Assignment: Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} 1& ,x≤ 0, y ...
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1answer
45 views

How to prove that $f_n(x) = \frac{1}{1+n^{2}x^{2}}$ is continuous on $[0,1]$?

I am having trouble verifying continuity. This seems like a very simple problem but I am not sure if my approach is correct: To prove that $f_n(x) = \frac{1}{1+n^{2}x^{2}}$ is continuous on $[0,1]$, ...
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$f :\mathbb R\to \mathbb R$ is a continuous function of period $1$ , then $f$ is uniformly continuous on $\mathbb R$

Let $f :\mathbb R\to \mathbb R$ be a continuous function such that $f(x+1)=f(x) , \forall x\in \mathbb R$ i.e. $f$ is of period $1$ , then how to prove that $f$ is uniformly continuous on $\mathbb R$ ...
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44 views

$V$ is open , then $V=\{x\in \mathbb R:f(x)>0\}$ for some continuous function $f$

Let $V$ be a non-empty open set of real numbers , then how to prove that there is a continuous function $f:\mathbb R\to \mathbb R$ such that $V=\{x\in \mathbb R:f(x)>0\}$
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123 views

Proving continuity by epsilon-delta proof for a function of two variables.

On account of a SE question , I raised the following question. Let $f:D \to \mathbb R^2$ be a function in two variables. How would we go about setting up an epsilon-delta proof? Let $f$ for example ...
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59 views

Prove a two variable function to be continuos on an specific domain

Let the function $g:D \to \mathbb R^2$ be given by $$g(x,y)=2-|x+y|$$ with domain $D= \{(x,y) \in \mathbb R^2:x+y \leq2\}.$ How to prove it is continuous? I know that I need to prove this for every ...
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29 views

Where does my proof of uniform continuity fail?

I am trying to prove that $f:R \to R f(x)=\sin x$ is uniformly continuous. I have said: Fix $\epsilon > 0$ and $\delta=\epsilon$ $|\sin x - \sin y| \le |\sin x| - |\sin y| \le 1 - 1 = 0 ...
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3answers
100 views

Continuous map from the ring on the unit circle

Is there a surjective continuous map from the ring $r<x^2+y^2<1\,(0<r<1)$ on the unit circle $x^2+y^2<1$ ? It seems NO, but how can it be done ? Edit: what if we add the ...
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2answers
37 views

$f$ is continuous and $f(V)$ is open whenever $V$ is open $\implies$ $f$ is monotone

Let $ A $ be a non-empty subset of $\mathbb R$ and $f : A \to \mathbb R$ be a continuous function on $A$ such that $f(V)$ is an open set for any open set $V$ , then how to prove that $f$ is monotone ? ...
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1answer
33 views

Proving a function is not uniformly continuous.

I am using the definition: $(∃ε > 0)(∀n ∈ N)(∃ x_n, y_n ∈ (0,1])[(|x_n − y_n| < δ_n =1/n) ∧ (|f(x_n) − f(y_n)| ≥ ε)]$ to prove that $1/x^2$ is not uniformly continuous. In the solution I am ...