# How to prove this function has Darboux's property? f ( x ) = { <mtabl

How to prove this function has Darboux's property?

How do I prove this function has Darboux's property? I know it has it because it has antiderivatives, but how do I prove it otherwise, with intervals maybe?
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Ordettyreomqu
Step 1
Take, $a,b\in \mathbb{R}$ with $a. You want to prove that, if y lies between f(a) and f(b), then there is some $c\in \left[a,b\right]$ such that $f\left(c\right)=y$. If $b<0$ or $a>0$, this is clear, by continuity. If $a<0, take some $n\in \mathbb{N}$ such that $\frac{1}{2\pi n}.
Step 2
Then $f\left(\left[\frac{1}{2\pi n+\pi },\frac{1}{2\pi n}\right]\right)=\left[-1,1\right]$, and therefore there is come $c\in \left[\frac{1}{2\pi n+\pi },\frac{1}{2\pi n}\right]$ such that $f\left(c\right)=y$. The remaining cases are similar.
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grenivkah3z
Step 1
It's obvious that it has Darboux's property on any interval [a, b] with $0.
If $a\le 0, then $f\left(a\right),f\left(b\right)\in \left[-1,1\right]$ anyways and there are some $0<{x}_{1}<{x}_{2} such that $f\left({x}_{1}\right)=-1$ and $f\left({x}_{2}\right)=1$. By the previous remark, every value between -1 and 1 (so a fortiori every value between $min\left\{f\left(a\right),f\left(b\right)\right\}$ and $max\left\{f\left(a\right),f\left(b\right)\right\}$) is attained in the interval $\left[{x}_{1},{x}_{2}\right]\subseteq \left(a,b\right)$.
Step 2
If $a<0\le b$, then $f\left(a\right),f\left(b\right)\in \left[-1,1\right]$ anyways and there are some $a<{x}_{1}<{x}_{2}<0$ such that $f\left({x}_{1}\right)=-1$ and $f\left({x}_{2}\right)=1$. For the same reason as the previous case, every value between -1 and 1 is attained in the interval $\left[{x}_{1},{x}_{2}\right]\subseteq \left(a,b\right)$.