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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continuity and limit of a function.

I am asking this question for explanation of what this question wants? To what degree would the sequence definition of continuity need to be modified in order to be suitable as a definition for ...
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0answers
23 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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0answers
28 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
25 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
78 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
88 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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1answer
46 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
30 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
25 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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0answers
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
21 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
51 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
47 views

Finding the values of $a$ and $b$ such that $f$ is continuous and differentiable at $x = 1$? [on hold]

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? [on hold]

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
23 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
29 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
11 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
35 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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0answers
16 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
50 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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0answers
38 views

Does differentiability imply continuity for a derivative? [closed]

If $f(x)$ is differentiable at a point $c$, then is $f'(x)$ continuous at $c$? If so (or if not,) please provide a proof or a counterexample.
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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
35 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
54 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
42 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 ...
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2answers
39 views

Continuous function vs Uniformly continuous function

Can you give me an example of the function in metric space which is continuous but not uniformly continuous. Definitions are almost the same for both terms. This is what I found on wiki: ''The ...
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2answers
71 views

What is wrong with this proof of continuity of a function of two variables?

If a function is define as: 1)$$f(x,y)=\begin{cases} \frac {2xy}{x^2+y^2} &\mbox{for} (x,y)\neq (0,0) \\0 &\mbox{for} (x,y)=(0,0) \end{cases} $$ Then the following proof argument, $$\frac ...
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1answer
37 views

No direct proofs of “if $ f: (X, d_X) \to (Y, d_Y)$ is continuous and $X$ is compact then $f$ is uniformly continuous.”

I am studying the theorem "if $f:(X,d_X)\to (Y,d_Y)$ is continuous and $X$ is compact, then $f$ is uniformly continuous." I am not looking for a proof, but I have an argument against any attempt at a ...
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2answers
58 views

If $g$ is a continuous function and $F(x) = x^3+ g(x)$, then $F$ is also continuous

I'm stuck with a problem, I'm not sure how to do solve it or what to do to make sense of it. It goes like this: $g(x)$ is a function that is continuous within $\mathbb{R}$ and $F(x) = x^3+ g(x)$ ...
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1answer
25 views

continuity of polynomial of two variables

We know that polynomial functions are always continuous. The proof which I did was only for single variable polynomial. What about the polynomials in two variables? Can we say a polynomial of two or ...
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1answer
40 views

On continuity of $f^{-1}$

Let $I$ be a non-empty real interval , then it is easy to prove that any injective continuous function $f:I\to \mathbb R$ is strictly monotone . Now let $A$ be a non-empty real set and $f:A\to \mathbb ...
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0answers
12 views

From essential oscillation to a continuous representative

Let $u$ be a measurable function such that for every(former: a.e.) $x\in \Omega$ there holds for sequences $R_n,\delta_n\to 0$ that$$\omega_n:=ess-osc_{B_{R_n}(x)} u\leq \delta_n. \tag{1}$$ Edit: ...
2
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2answers
120 views

Let $f:[a,b] \rightarrow R$ be continuous such that $f(a)=f(b)$.

Let $f:[a,b] \rightarrow R$ be continuous such that $f(a)=f(b)$. Show that for each $\epsilon>0$, there exist distinct $x,y \in [a,b]$ such that $|x-y|<\epsilon$ and $f(x)=f(y)$. I can not ...
2
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1answer
43 views

Derivative of an Inverse Function

Can someone please give me a simple proof of this- If $f$ is differentiable on an interval containing $c$ and $f'(c) \neq 0$, then $f^{-1}$ (inverse of $f$) is differentiable at $f(c)$. I can see ...
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1answer
43 views

Does $f_n(a_n)\to f(a)$ hold?

Say, we have $f, f_n \in C^0(\mathbb R, \mathbb C)$ such that $f_n \xrightarrow{\text{uniform}}f$ and a sequence of reals $a_n \to a$. Does it then hold that $f_n(a_n)\to f(a)$? I couldn't think of ...
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1answer
31 views

Continuity of f [closed]

Is the statement below true.If it is could someone provide a proof of this.If its not provide a counter example $ f(x)$ is continuous at $x_0$ $\implies \exists \delta>0:$ (if ...
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1answer
47 views

Question about limit points in relation with continuity and functional limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have the feeling that the author is being careless about limit points in his theorems or I am not understanding ...
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2answers
69 views

A continuous mapping $f:\mathbb{R}\rightarrow\mathbb{R}$ may have a fixed point?

Let a function $f:\mathbb{R}\rightarrow\mathbb{R}, $satisfied $$\forall x,y\in\mathbb{R},|f(x)-f(y)|\leq k|x-y|.(0<k<1)$$ Prove: There exists a only one $\xi\in \mathbb{R}$ ,such that ...
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2answers
83 views

Characterization of continuity in terms of preimages of open sets

1--8 Theorem. If $A\subset \mathbb R^n$, a function $f:A\to \mathbb R^m$ is continuous if and only if for every open set $U\subset \mathbb R^m$ there is some open set $V\subset \mathbb R^n$ such ...