# Tagged Questions

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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### A Topology such that the continuous functions are exactly the polynomials

I was wondering which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous if and only if it is a polynomial function $f(x)=a_nx^n+\cdots+a_0$. Obviously, the ...
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### Why did mathematicians introduce the concept of uniform continuity?

I have solved many problems regarding uniform continuity, but still I can't understand the following: Is there any practical application of this concept, or it is just a theoretical concept? Is there ...
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### Why are real numbers useful?

A question (by a fellow CS student taking a first course in calculus, presumably after the lecture in which continuity was introduced: was as follows. In the real, physical world, we deal with ...
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### What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference geometrically? What is the best way to describe the difference between these two concepts? Where the motivation of ...
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### Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? It seems to me like they are equal definitions in a way. Can you give me a counter-example? Thanks
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### Do all continuous real-valued functions determine the topology?

Let $X$ be a topology space. If I know all the continuous functions from X to $\mathbb R$, will the topology on $X$ be determined? I know the $\mathbb R$ here is, somewhat, artificial. So if this is ...
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### Can a continuous function from the reals to the reals assume each value an even number of times?

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. Is it possible for $f$ to assume each value in its range an even number of times? To clarify, some values might be taken 0 times, some 2, ...
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### Is some thing wrong with the epsilon-delta definition of limit??

In the epsilon-delta definition of limit which is: For all $\epsilon>0$ there exists a $\delta>0$ such that, whenever $|x-a|<\delta$ then $|f(x)-L|<\epsilon$ . Now since $\epsilon$ ...
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### $f(f(\sqrt{2}))=\sqrt{2}$ then f has a fixed point

$f(x)$ is continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ $f(f(\sqrt{2}))=\sqrt{2}$ Prove that $f$ has a fixed point in other words prove the there is $x_1$ such that $f(x_1)=x_1$ I tried using ...
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### Is the plane minus a line segment homeomorphic with punctured plane?

Is $\mathbb R^2$ minus a line segment i.e. $\mathbb R^2 \setminus ([0,1]\times \{0\})$ homeomorphic with a punctured plane $\mathbb R^2\setminus \{(0,0)\}$ ?
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### Condition for an additive function to be continuous

The problem below is Problem 7 from this year's Miklos Schweitzer competition (contest ended Nov 4th). Suppose that $f: \Bbb{R} \to \Bbb{R}$ is an additive function (that is $f(x+y) = f(x)+f(y)$ ...
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### If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$?

Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ?
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### Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty)$ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
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### Who gave you the epsilon?

Who gave you the epsilon? is the title of an article by J. Grabiner on Cauchy from the 1980s, and the implied answer is "Cauchy". On the other hand, historian I. Grattan-Guinness points out in his ...
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### Continuity of a function in two variables

Function $f(x,y)$ is continuous in each variable separately. Prove that there exists a point where it is continuous in two variables. I do not quite understand how to act here. I know the definition ...
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### $f$ be a smooth function on real line , $f(0)=0$ , $f(x)>0, \forall x \ne 0$ and any $f^{(n)}(0)=0$ ; is $\sqrt f$ smooth?

Let $f: \mathbb R \to \mathbb R$ be an infinitely differentiable function such that $f(0)=0$ , $f(x)>0 , \forall x \ne 0$ and $f^{(n)}(0)=0$ ( the $n$-th derivative ) $, \forall n \in \mathbb N$ ...
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### 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 ...