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The function tan(x)=x has 1. Unique fixed point 2. No fixed point 3. Infinite many fixed points 4. More than one but finitely many fixed points.

My attempt: I have solved this problem by graphical approach. The intersecting points of the function y=tan(x) and the line y=x are fixed points that I obtained which are infinite. But my problem is that how to solve it without graphical approach and how to discard options. Is there any general approach to tackle this type of problems?

Mathforjob
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    There's no general approach. It all depends on the function. Sometimes, setting $f(x)=x$ leads to a nice algebraic solution, or sometimes rote argumentation is needed. Other times, a combination of the two works. In your case, I would simply argue that since $\tan(x)$ takes on all real values continuously on the domain $(n\pi-\frac\pi2,n\pi+\frac\pi2),;n\in\mathbb Z$, that shows that there is exactly one fixed point per value in $\mathbb Z$. – Rushabh Mehta Dec 24 '18 at 22:23
  • If x is non negative real number and it is not an odd multiple of pi/2 then what about fixed points. – Mathforjob Dec 24 '18 at 22:45

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