How to prove this function has Darboux's property?

$f(x)=\{\begin{array}{ll}\mathrm{cos}(\frac{1}{x})& \text{if}x\ne 0\\ 0& \text{if}x=0\end{array}$

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?

$f(x)=\{\begin{array}{ll}\mathrm{cos}(\frac{1}{x})& \text{if}x\ne 0\\ 0& \text{if}x=0\end{array}$

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?