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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Where is $f(x)=\sqrt{|1-x^2|}$ Lipschitz continous?

It seems to me that the Lipschitz constant is 1 near $x=\pm 1$, $y= \pm 1$ $$ |f(x)-f(y)| \leq \frac{|x+y|}{\sqrt{|1-x^2|}+\sqrt{|1-y^2|}}|x-y| $$ How would you define the Lipschitz constant L?
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1answer
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continuity at a point

If $f:\mathbb R^2\to \mathbb R$ is given by $f(x,y) = \left\{ \begin{array}{cc} [\frac{\sin x}{x}]+[\frac{y}{\sin y}] & \mbox{if } xy \neq 0 \\ 2 & \mbox{if } xy = 0 \end{array} \right.$ ...
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33 views

Mean value theorem of a function in [a,b]

Is Mean Value Theorem (Rolle's Theorem) applicable for the following function: $$\log \frac{x^2 + ab}{(a+b)x}$$ in the interval $(a,b)$ My text says that it's applicable. But isn't the function ...
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44 views

continuity of delta function [on hold]

is the diract delta function continous?
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2answers
60 views

Is every convex function on an open interval continuous?

Let $f:(a,b)\rightarrow \mathbb{R}$. $f$ satisfied the following property: If $\forall x_{1},x_{0},x_{2}\in(a,b)$ and $x_{1}<x_{0}<x_{2};$then$\frac{f(x_{0})-f(x_{1})}{x_{0}-x_{1}}\geq ...
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1answer
17 views

vector function and scalar function?

If we have $\lim_{{\bf x}\to {\bf 0}} f({\bf x})=L$, where ${\bf x}$ is a vector and we know that ${\bf x}=(x_{1},\dots , x_{n})$ can we instead write that $\lim_{{\bf x}\to {\bf 0}} f(x_{1},\dots ...
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1answer
32 views

What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
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21 views

About level curves of a continuous function in a real square, and connectivity

Assume f is a continuous function on the (unit) square in real plane. Name the edges N,S,E and W in the natural way. Assume f is >0 at W edge and <0 on E edge. Intuitively it is clear that there ...
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55 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
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2answers
45 views

Finding limits using $\epsilon$ - $\delta$ method

How do I prove the following function is continuous at $(0,0)$ using epsilon-delta approach? $$\frac{x^5-y^5}{x^2+y^2}$$
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2answers
52 views

Does intermedia value theorem apply to continuity on open intervals?

Does the intermediate value theorem apply to functions that are continuous on the open intervals (a,b)? I know it's pretty vital for the theorem to be able to show the values of f(a) and f(b), but ...
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7 views

Continuity of a function, after extending by oddness.

I'm blanking on how to prove a claim my professor stated in class, even though it should be really simple. We're working on deriving the homogeneous wave equation with clamped boundary conditions. So ...
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3answers
53 views

Prove this function is pointwise continuous

Prove that the function $f\colon(0,1)\cup(1,2)\mapsto\mathbb{R}$ is continuous at all points in its domain. $$f(x)= \begin{cases} x : 0<x<1\\ 0 : 1<x<2 \end{cases}$$ The graph of $f$:
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2answers
41 views

An intuitive idea about the limit of a continuous function: Is it correct?

Let's assume that we have a function $f(x)$ whose limit at $c$ is given as $\lim_{x \to c}f(x) = f(c)$, such that it is continuous at $c$. For this limit, we have left and right side limits $\lim_{x ...
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2answers
41 views

Counterexamples for $f(\overline{A}) = \overline{f(A)}$ and $\overline{f^{-1}(B)} = f^{-1}(\overline{B})$ in (non-)continuous mapping $f: X \to Y$

Let $f$ be a mapping. Prove that the following three statements are equivalent. $f$ is continuous; $\forall A \subseteq X: f(\overline{A}) \subset \overline{f(A)}$; $\forall B \subseteq ...
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2answers
55 views

continuity and limit of a function.

Below is the question: To what degree would the sequence definition of continuity need to be modified in order to be suitable as a definition for the limit of a function? In other words,if ...
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25 views

Basic calculus question with continuous function [duplicate]

This is actually not my question, it was asked yesterday by user176744 in this link $[0,n]$ continuous function problem and I feel as if it didn't get enough attention. I am also interested in a ...
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31 views

Condition for a function $f: \mathbb R \rightarrow \mathbb R$ being right or left-continuous at $a \in \mathbb R$.

I know that $f: X \rightarrow \mathbb C$ is continuous if and only if for every convergent sequence $(x_n)$ in $X$ the identity holds $\lim_{n \rightarrow \infty} f(x_n) = f(\lim_{n \rightarrow ...
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1answer
26 views

If $f$ is $C^1(\mathbb{R})$, is it $C^1(\{a\})$?

Say I have a well-behaved function like $f(x)=x$. This is obviously $C^1$, but does it make sense to say the function is $C^1$ around a single point? A broader question, if $a\in\mathbb{R}$, does ...
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1answer
86 views

Let $f:(\mathbb{R}\setminus\mathbb{Q})\cap [0,1]\to \mathbb{Q}\cap [0,1]$. Prove there exists a continuous$f$.

I'm working on the following problem from N.L. Carother's Real Analysis: Let $I=(\mathbb{R}\setminus\mathbb{Q})\cap [0,1]$ with its usual metric. Prove that there is a continuous function $g$ ...
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5answers
96 views

Definition of continuity

It has been a year or so I took my course of real analysis, still could not understand these two definitions of continuity-(These two definitions are given as chapter 9 and 10 in the classroom ...
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56 views

Suppose all partial derivatives of $f$ exist at $x_0$; is $f$ continuous at $x_0$?

Consider $f : C \to \mathbb{R}$ with $C \subset \mathbb{R}^n$ being open: Suppose $f$ is differentiable at $\mathbf{x}_0 \in C$. Is $f$ continuous at $\mathbf{x}_0$? Why? Suppose all partial ...
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1answer
31 views

Why are the following graphs discontinuous at $f(0)$ (epsilon-delta)

The caption for graph (f) is "Infinite jump". The caption for graph (h) is "Infinitely many infinite jumps". The graphs are meant to illustrate that we can pick arbitrarily small intervals around ...
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2answers
27 views

On the definition of Scott Continuity

I somewhere encountered the concept of "Scott Continuity" as follows. Let $P,Q$ be partially ordered sets; a function $f:P\to Q$ is Scott continuous if it preserves directed suprema, i.e. for all ...
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9 views

Sobolev spaces and Lipsschitz continuity [duplicate]

How to show that u $\epsilon$ ${W^{1,\infty}(\Omega)}$ if and only if u is Lipschitz continuous. But I suggested to use the fact that u is Lipshtz means that there is a constant $L>0$ such that ...
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0answers
22 views

On a step in a solution of a proof of the Continuity of thomae's function at irrationals.

Thomae's function: $$t(x) = \begin{cases} 0 \ \ \text{if} \ x \in \mathbb{R}- \mathbb{Q} \newline \frac{1}{q} \ \ \text{if} \ x \in \mathbb{Q} \ \text{and} \ x = \frac{p}{q} \ \text{in lowest terms} ...
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4answers
53 views

If $f$ is continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ and if $lim_{x\to\infty}f'(x)=0$ Then $f$ uniformly continuous on $[0,\infty)$

I got this problem: Let $f$ be a continuous function on $[0,\infty)$ and differentiable function on $(0,\infty)$ such that $\lim_{x\to\infty}f'(x)=0$. (1) Prove that for each $0<\epsilon$ there ...
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1answer
21 views

Violation of IVP of a continuous functions

We know that IVP of a continuous function says that if $f:\mathbb R\rightarrow \mathbb R$ be a continuous function on $\mathbb R$ then between $[a, b]$ there will be at least one real root of ...
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3answers
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Finding the values of $a$ and $b$ such that $f$ is continuous and differentiable at $x = 1$? [closed]

The equation is $F(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ ax+b & \text{if } x>1 \end{cases}$ Differentiable at $x = 1$ I'm having a hard time understanding on how to ...
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1answer
27 views

Does a continuous function map a countable dense set to a countable dense set? [closed]

Let $(\mathscr{X}_i, d_i), i =1,2$ be metric spaces. Let $f$ be a continuous function from $(\mathscr{X}_1,d_1)$ to $(\mathscr{X}_2,d_2)$. If $D \subset \mathscr{X}_1$ is countable and dense, is ...
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1answer
28 views

Continuity and differentiability on piecewise function

Let $$f(x)=\begin{cases}x^2-3, & x<0;\\-3, & x\geq 0.\end{cases}$$ (a) Find the value of $x$ where $f$ is discontinuous (b) Find the value of $x$ where $f$ is non-differentiable ...
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3answers
31 views

Show a continous function is bounded on a closed interval

For a homework problem, I need to show a function $\pi + 0.5\sin(\frac{x}{2})$ is bounded on the interval $[0,2\pi]$. I'm having trouble conceptualizing a good way to do this though. Can anyone help? ...
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1answer
13 views

Is the principal value of Argument differentiable at every nonnegative nonzero number?

How do i show that argument is continuous at points except its branch cut? I posted a question to ask whether the principal value of Argument $Arg:\mathbb{C}\setminus \{0\}\rightarrow (-\pi,\pi]$ is ...
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1answer
36 views

Convergence of a sum of sines

If $ s_N(x) := \sum_{n = 1}^N c_n \sin(n x) $ converges uniformly on $[0, \pi]$ as $N \to \infty$ then $c_n = o(n^{-1})$. a) Is $c_n = o(n^{-1})$ sufficient for uniform convergence? b) Is $\sum_n n ...
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1answer
72 views

Show that f is onto.

Let $X$ be a compact connected Hausdorff space and $f:X\rightarrow X$ a continuous open map. Show that f is onto.
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1answer
51 views

continuity extension of exponential $f(x)= a^x$

Consider tha exponential function $f(x) = a^x$, where $f: \mathbb{Q} \to \mathbb{R}$. My problem is to show that it has unique extension and how am I going to define this one? Also, I used a ...
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17 views

Comparing notions of continuity

I have trouble distinguishing 3 different types of continuity Uniform continuous Sequential continuous Equicontinuous Could someone explain the difference between 3 and give some examples? I am ...
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1answer
52 views

A Question on continuity of a piecewise function

I wanted to know, how to check the continuity at $(0,0)$ of the following function: $ f(x,y)= \begin{cases} \frac{x^2y^2}{x^3+y^3} & \text{$x^3+y^3\ne0$}\\ 0 & ...
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0answers
31 views

Extending linear continuous functions.

Let $E$, $F$ be normed vector spaces and $M$ a subspace of $E$. I'm trying to find an example of a function $f:M\to F$ such that $f$ is linear and continuous but that you can't extend it to ...
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1answer
25 views

Discontinuity of a piecewise defined function with a parameter

Let $$ f(x,y) = \left\{ \begin{array}{ll} cx+4, & \textrm{if $x<6,$}\\ cx^2-4, & \textrm{if $x\geq 6.$}\\ \end{array} \right. $$ respectively. For what value of $c$ is this function ...
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1answer
38 views

Is it true that $\textrm{supp}(f)\subseteq K$ implies $f|_{\partial K}=0$?

Maybe this will be an elementary question but I need to clarify this. Let $X$ be a metric space and let $f:X\longrightarrow \mathbb R$ continuous. Suppose $\textrm{supp}(f)\subseteq K$ where $K$ is ...
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0answers
28 views

Does continuity follow from linearity on all or only finite-dimensional vector spaces

I'm currently reading an introduction book on topology. While solving one of its exercises I came across something odd. The exercise is: Let $E$ and $F$ be normed spaces, let $T:E \to F$ be linear, ...
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2answers
17 views

approximate a Borel set by a continuous

I wonder if it is possible to approximate a Borel set by a continuous function i.e. Let $B$ a Borel set in $(X,d)$ (compact separable metric space) I wonder if there continuous functions ...
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0answers
10 views

Integration of a continuous function under Lebesgue-Stieltjes measure space using simple functions

I am struggling to prove the following result using an approximating sequence of simple functions. Could anyone give me a clue? Under a Lebesgue-Stieltjes measure space ...
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2answers
44 views

Proving the existence of fixed point $\alpha \in [-1,1]$

Can anyone help me with the following problem: I don't have the slightest idea on where to start: Consider a function $g$ which is continuous on the compact interval $[-1,1]$ such that: $g(-1)=0$, ...
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1answer
36 views

Relationship between Continuity and Countability

This is a consequence of one of the problems in elementary real analysis that I am attempting to solve. I have this doubt. Suppose $f$ is a continuous map from the reals to the reals. If the set ...
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1answer
39 views

For every intermediate value, there exists a sequence that converges to it.

I want to prove that: If the continuous function $f(x)$ has a bounded limt as $x$ goes to $\infty$ i.e $$0<L=\liminf (f)\leq S=\limsup(f)<\infty,$$ then for every $x_0 \in [L,S]$ there ...
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1answer
36 views

Nowhere Continuous Function [duplicate]

I was reading Dirichlet and Thomae's functions and got interested to know about functions which are continuous nowhere. Since these have a lot to do with rationals and irrationals, the next question ...
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1answer
55 views

Prove $p_k\circ f$ continuous $\implies$ f is continuous

Let $X_1,\dots X_n$ topological space and $p_k:X_1\times\cdots X_n\to X_k$ the projection to the kth component. Let $Y$ be topological space and $f:Y\to X_1\times\cdots\times X_n$ function s.t ...
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1answer
43 views

For continuous functions, preimage of open set is open.

Let $f$ be a continuous function from a metric space $X$ into $Y$. If $V\subset Y$ and $V$ is open, then show that $f^{-1}(V)$ is open. The proofs I've seen of the fact that open sets have open ...