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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Comparing smoothness among approximations

We are interpolating a missing fragment of a 2D curve given a set of sample points. Our method generates several candidates of curve pieces to fill the missing part, but we want to select the solution ...
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Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with "one-sided" limits or derivatives. This is something I find ...
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34 views

Prove $\{f_n(x)\}$ is not continous

$f_n(x) = \left\{ \begin{array}{ll} \frac{1}{n} & \quad x \in \mathbb{Q} \\ 0 & \quad x \notin \mathbb{Q} \end{array} \right.$ Not sure how to show ...
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41 views

How can one prove that a real function is closed? [closed]

I am defining a closed function to be one that takes closed sets to closed sets. Given a function, domain and codomain, you could prove that it is not closed by simply providing a counter example ...
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Preimage of sets, complement of sets, continuity of functions

I just got some simple questions in real analysis regarding preimage and complement of sets and continuity. Suppose $f:X\to Y$, then does $f^{-1} (Y\setminus F)=f^{-1} (Y)\setminus f^{-1} ...
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Example of a function that converges to 0 pointwise but integral is 3/2?

Give an example of a sequence of continuous functions $(f_n)$, $f_n : [0, 1] \to \mathbb{R}$ that converges to zero pointwise, and such that the integral of each function within the given domain is ...
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146 views

Uses of step functions

My highschool teacher has informally told us about what continuity is and used step functions as an example of a discontinuous function. The Wikipedia page for it links to a lot of other kind of step ...
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Let $f$ be a real-valued continuous function on $[0,1]$ which is twice continu-ously differentiable on $(0,1)$. Suppose that $f(0) = f(1) = 0$

Let $f$ be a real-valued continuous function on $[0,1]$ which is twice continu-ously differentiable on $(0,1)$. Suppose that $f(0) = f(1) = 0$ and $f$ satisfies the following equation: $$x^2f''(x) + ...
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33 views

Continuous function on a non-compact set

I'm trying to show if $X$ is non compact ($X \subseteq \mathbb{R}$) then there is a cont function $f:X \rightarrow \mathbb{R}$ which is bounded but doesn't attain it's bounds. I'm trying it for a set ...
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34 views

Explain how L(g,P) = U(g,P) implies continuity of g.

First, let $g$ be bounded on $[a,b]$. Now, assume $\exists P$, a partition, such that $L(g,P)=U(g,P)$. I am told the correct answer to the question "describe $g$" is that $g$ is continuous on ...
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25 views

Function of metric with a fixed point

I'm trying to prove that given a metric space $(X, d)$, for a fixed $x\in X$, define the function $g(y)=d(x,y)$, then $g(y)$ is continuous, using triangle inequality. My first question is that can I ...
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38 views

What is an example of a continuous but not closed function? [duplicate]

I have two questions about closed functions. Firstly, we say that a function is closed if it maps closed subsets in the domain to closed subsets in the co-domain. Polynomials are typical examples of ...
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30 views

continuity of a piece wise function defined partially on a closed interval

using epsilon delta definition prove that $f(x)=\left \{ \begin{array}{cc} 2 & : x \in[0,1]\\ 1 & : x=-1 \end{array}\right.$ is continuous on $E= [0,1] \cup \{-1\}$. Here is my attempt. I ...
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34 views

A smooth function which is nowhere real analytic, and preserves rationality of its argument

There are examples $\!^{[1]}$$\!^{[2]}$ of continuous infinitely differentiable (class $C^\infty$) functions $\mathbb R\to\mathbb R$ that are nowhere real analytic. I wonder if it is possible to ...
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16 views

Is $f(x,y) = (x^2-y^2,xy)$ lipschitz on $\mathbb{R}^2$?

How can I show that the $f$ is lipschitz? I try to calculation such that $|f(x_0, y_0) - f(x_1,y_1)|^2 = ((x_0^2 -y_0^2)-(x_1^2 -y_1^2))^2 +(x_0 y_0 -x_1y_1)^2$ and $|(x_0-x_1,y_0-y_1)|^2 = ...
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15 views

Volume Zero of Not Continuous Function

Show that a bounded real-valued function f on a closed interval $I$ of $E^n$ is integrable on $I$ if and only if the set of points of $I$ at which $f$ is not continuous is the union of a sequence of ...
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23 views

Continuity of a map in a metric space

Let $C^0([a,b])$ denote the space of continuous function $f:[a,b]→\Bbb R$. Define $ d(f,g)= \sup_{[a,b]}|f-g| $. We define $F:C^0([a,b])→\Bbb R$ to be $F(f)=\int_a^b f$. I want to show that $F$ is a ...
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27 views

Is there a way to calculate RMS value continuously?

Using that the RMS by definition is: $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ which can be calculated by using Riemann sums in the following way: $\sqrt {\frac 1N\sum_0^Nf[i]^2} $ I've tried that in ...
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1answer
25 views

Mean value theorem, Wierstrass theorems

I have a question that is related to these theorems I tried to tackle but got stuck Please let me know if it is the proper way to go ? The question is: Let $f:[0,1]\rightarrow\mathbb{R}$ be a ...
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48 views

Metric space and continuity

We define a map $f:(S,d)→(S',d')$ between 2 metric spaces to be continuous at x belongs to S if for every sequence ${x_n}$ in $S$ that converges to x, the sequence {f(x_n)} in $S'$ is convergent to ...
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83 views

Showing a particular type of continuous function is uniformly bounded

Let $I = [0,\infty)$ and $f:I \to I$ be continuous with f(0) = 0. Show that if \begin{equation} f(t) \leq 1 + \frac{1}{10}f(t)^2, \text{ for all } t \in I \end{equation} then $f$ is uniformly bounded ...
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26 views

Proof of Interceting Lines

I have this practice problem from a final exam study guide. Let $f,g$ be continuous on $[a,b]$ and $f(a)>g(a)$ but $g(b)>f(b)$. Prove that $\exists c \in [a,b]$ such that $f(c)=g(c)$. My ...
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40 views

Complex supremum function is strictly monotone

I'm having great troubles to solve the following exercise: Let $f$ be a holomorphic function on the unit disc. For $0\leq r < 1$ is $$M(r):=\sup\limits_{|z|=r}|f(z)|$$ Show that ...
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28 views

Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C_b[0,1]$

Following Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ I would like to prove that the same is true for bounded functions on $[0,1]$ ...
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36 views

Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.

I have some questions about this proof that "Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.": By the example (12), we just have to consider the ball $B(0,1)$, we ...
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18 views

How is $d(af(x), af(x_o))$ and $d(f(x), f(x_o))$ related?

I wish to prove that given $f \in C_0([0,1])$ of continuous function, then $af \in C_0([0,1])$ where $a \in \mathbb{R}$ I am having trouble relating $d(af(x), af(x_o))$ with $d(f(x), f(x_o))$ So to ...
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55 views

Prove that $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ is continuous and can be differentiated ad infinitum

We have $f:(0,\infty) \rightarrow \mathbb{R}$ defined by infinite series $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ Prove that $f$ is continuous and can be differentiated ...
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derivative of differentiable function [duplicate]

Edited: It is known that if $f$ is differentiable then the derivative function of $f$ is not always continuous. For instance $f(x)=x^2\sin (\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$ if $x=0$. Then ...
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$f \in C^1[0,\infty)$ such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then for $s<2$ ; $\lim_{x \to \infty}x^{-s}f(x)=\infty$?

Let $f \in C^1[0,\infty)$ be such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then is it true that for $s<2$ , $x^{-s}f(x) \to \infty$ as $x \to \infty$ ?
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How can I write a gradient of sobel filter in continuous formula?

Let $*$ denote a convolution operation, $G$ denote a kernel, and $I$ is a given image. The gradient of the image $I$ is equivalent: $\nabla (G*I) = (\nabla G) * I$ The Sobel filter approximtes two ...
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The continuity of injectivity radius

Let $M$ be a Riemannian manifold. $r:M\to [0,+\infty]$ denotes the function assigns to $p\in M$ the injectivity radius $r_p$ of the exponential map $\exp_p$. Is this function $r$ is continuous or ...
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Why is continuity permissible at endpoints but not differentiability?

Differentiable at endpoints? cause of differentiation only on an open set. Admittedly, there are some questions and answers as to why a function defined on a closed interval is not differentiable on ...
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21 views

For which $r$ element of $(0,\infty$) $|x|^r \to 0$ if $x \to 0$ [closed]

For which $r$ element of $(0,\infty$) $|x|^r \to 0$ if $x \to 0$? Somehow I have no idea how to solve the problem. I know that it's true for $r\geq 1$.
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77 views

Prove that $a^x$ is continuous for a>0

Here is what I need to prove: Let $a>0$ be a positive real number. Then the function $f: \Bbb R\to \Bbb R$ defined by $f(x):=a^x$ is continuous. We are not supposed to use logarithms. Some of ...
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43 views

Continuous and Inverse function

I need to prove that if $X$ is a subset of $\mathbb{R}^n$ and $Y$ is a subset of $\mathbb{R}^m$, and $X$ and $Y$ are closed and bounded, then if $f:X \rightarrow Y$ is continuous and has a inverse ...
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51 views

Intermediate value property under some conditions imply continuity

Let $f\colon \mathbb{R} \to \mathbb{R}$ be a function with the intermediate value property. Let $x \in \mathbb{R}$. Suppose to each sequence $ (x_n) $ converging to $x$ there exists a constant $M$ ...
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69 views

How to define $[-\infty, \infty]$ or $[0, \infty]$?

I am familiar with basic undergraduate topology. For example, I know the process of one point compactification of a non-compact topological space, and how it applies to, say, $\mathbb R^2$. My ...
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To show that for every continuous function there exists some other continuous function satisfying this conditions

Suppose that we start with some continuous function $f$ defined on $[a,b]$. Since it is continuous it is integrable so the number $\int_{a}^{b}f(x)dx$ exists. How to show (in an as elementary as ...
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19 views

Prove continuity of a function from $S^n$ to $\Bbb R^n$

Given an interval $k$-coloring of $[0,1]$, define a function $f: S^k \to \Bbb R^k$ as follows ($S^k$ is the $k$-sphere). Let $x = (x_1,x_2,...,x_{k+1})$ be a point on the $k$-sphere $S^k$. Define $z = ...
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35 views

Are there continuous bijective functions from $\mathbb{R}$ to $\mathbb{R}$ with the standard topology that are not open functions?

Both domain and codomain with the standard topology, I can't imagine any example, could somebody give me one if it exists? I apologize if this was asked before, I didn't found it. In any case if this ...
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28 views

Continuity and uniform convergence counterexample

I have been having trouble finding a suitable counterexample to my problem, which I have written below. For each $n\geq 1$, let $f_{n}\colon \mathbb{R}\to\mathbb{R}$ be a continuous function and ...
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45 views

Let $g(x) = \sqrt[3]{x}$

a) Prove that $g$ is continuous at $0$ This is fairly straight forward $ \forall \epsilon >0$ $\exists \delta >0$ s.t. when $|x-c|<\delta$, it follows that $|f(x)-f(c)|<\epsilon$ At ...
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24 views

Example of convergent sequence and discontinuous function

I need a counter example to show the following statement is false: A function $f$ is continuous at a point c if there exists a sequence $x_n$ such that $x_n \rightarrow c$ as $n \rightarrow \infty$ ...
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Prove of the equivalence of semi continuity and open function

I'm totally stuck at this question. Somebody please help me~~!! Let X and y be nonempty subsets of $R^{N}$ and $R^{K}$ respectively. For a correspondence $F: X \rightarrow Y$ and $B \subset R^{K}$, ...
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22 views

Prove of continuity and open set

I need to prove this but I can't figure out how. It would be nice if somebody can help me out with this . Let X and Y be nonempty subsets of $R^{N}$ and $R^{K}$, respectively. Prove the followings: ...
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6 views

Proof of set being continuous

While I was doing my homework, I'm stuck at this question. Can somebody please explain me how this works?? Define the correspondence $B: R_{++}^{L} \times R_{++} \rightarrow R_{+}^{L}$ by $B(p,w) ...
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24 views

How to discuss the continuity and differentiability of $f(x)$?

I have this problem on my Real Analysis problem set: Let $I_{A}(x)$ be the characteristic function of any set A. Consider $\begin{cases} f(x) = x^2 I_{\mathbb{Q}}(x)\\ g(x) = x^2 I_{\mathbb{R - ...
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13 views

Differentiatiable functions question

Suppose that $f:(0,∞)↦(0,∞)$ is any differentiable function with the property that $f(\frac{1}{x})=f(x)$ for all $x\in (0,∞)$. Show that $f'(1)=0$ Honestly don't even know where to begin with this ...
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37 views

Semicontinuity of stochastic kernel

Let $X$ and $Y$ be metric spaces with Borel sigma algebra and $P(B|y)$ be a stochastic kernel on $X$ given $Y$. I'm trying to proof the equivalence of the following two statements: (i) $P(\cdot,y)$ ...
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19 views

Continuity of the function $F(x)=\int_0^{+\infty} f(x,t) d t$

Let $f(x,t): \mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$ be a smooth function (that is any order of derivative with respect $x$ and $t$ exists). Can we say $$F(x)=\int_0^{+\infty} f(x,t) d t$$ ...