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Is it possible to find non constant, continuous simple function ? While calculating Lebesgue integral of $f(x) =x^{2}$ on $ \mathbb R $. We have constructed simple function which is basically function with finite range . I was wondering will there exist any simple function which is continuous and non constant on any arbitrary space

Falcon
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SHREYA PANDEY
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    Simple functions take a finite number of values and cannot be continuous. –  Nov 30 '20 at 20:27
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    What about in general space (topological space... ). It has to continuous then each connected component of domain but it is not possible in R. – SHREYA PANDEY Nov 30 '20 at 20:35
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    Continous simple functions must be constant at each connected component of the domain. If the domain is $\mathbb{R}$ (which is connected), the continuous simple function has to be constant. – Ramiro Dec 01 '20 at 00:37
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    Ok, but does exist any other space where it is possible. – SHREYA PANDEY Dec 01 '20 at 05:01

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Proposition. A space $X$ admits a non-constant continuous simple function iff $X$ is not connected.

Proof. $(\Rightarrow)$ Let $f:X\to\Bbb R$ be a non-constant continuous simple function. Pick any $y\in f(X)$. It is easy to check that $f^{-1}(y)$ and $X\setminus f^{-1}(y)$ is a partition of the space $X$ into its disjoint non-empty open subsets.

$(\Leftarrow)$ If $X$ is a union of two its disjoint non-empty open subsets $A$ and $B$ then the characteristic function of the set $A$ is a non-constant continuous simple function on $X$. $\square$

Alex Ravsky
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