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Let $f : \mathbb R \to \mathbb R$ be a function. $f$ is said to have intermediate value property if $f$ takes connected subsets of real line to connected subsets of real line i.e. if $a,x,y \in \mathbb R$ are such that $f(x) < a <f(y)$, then $f(z)=a$ for some $z$ in between $x$ and $y$.

My question is : If $f: \mathbb R \to \mathbb R$ satisfies intermediate value property , then for every integer $n \ge 2$ , does $ f_n :\mathbb R \to \mathbb R$ defined as $ f_n (x)=(f(x))^n,\forall x \in \mathbb R$ , also satisfies intermediate value property ?

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Notice that if $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ satisfy the intermediate value property, then so does $g\circ f$.

Now take $g(x)=x^n$.

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