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Theorems for Continuity

Let fx=[x] ...

Question

Let $f (x) = [x]$ and $g (x) = {\begin{matrix} 0, x \in Z x^{2}; x \in R - Z \end{matrix}$ , then $g \circ f$ is continuous for $x \in A$ . (where $[.]$ denotes greatest integer function), find $A$ ?

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Solution

As image set of [.] (G.I.F.) function
only contains integers and g(n)=0
when $x ϵ z$ , therefore, go f(x)=0 for
$\forall x ϵ R \Rightarrow A \equiv R$ (set of real numbers)

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