# 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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### $f:[0,1] \to [0,1]$ be continuous bijection , $g \in C[0,1]$ and such that $\int_0^1g(x)(f(x))^{6n}dx=0, \forall n\ge 0$ , then $g=0$?

Let $f:[0,1] \to [0,1]$ be continuous bijection , $g:[0,1] \to \mathbb R$ be continuous such that $\int_0^1g(x)(f(x))^{6n}dx=0, \forall n\ge 0$ , then is it true that $g(x)=0,\forall x \in [0,1]$ ? ...
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### Showing $(C[0,1],d_{\infty})$ is connected

Prove that the metric space $(C[0,1], d_{\infty})$ is connected. Is it path connected? I know how to typically show that a set is connected, but to show $C[0,1]$ is connected is currently escaping ...
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### Holder continuous but not Lipschitz

Is there a function that is Holder continuous but not Lipschitz continuous?
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### Showing the continuity of $d(x,f(x))$

Assume that $(X,d)$ is compact, and that $f: X \to X$ is continuous. Show that the function $g(x) = d(x,f(x))$ is continuous and has a minimum point. Consider the function $g(x) = d(x,f(x))$. If $g$ ...
### How is the function $f: \mathbb{Z} \to \mathbb{R}$ continuous?
Where $\mathbb{Z}$ is the set of integers and $\mathbb{R}$ the set of real numbers. In a question in a problem sheet, it said this statement was correct, however I do not understand how. You ...
Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove ...