Math Problem

Ok I am so lost on where to even start on this problem. My teacher doesnt use a book and I havent been able to find anything on google to help me so I've turned to Anon to help me out.

draw a graph of a continuous function f from the closed interval [0,7] to [0,7] that satisfies all of these properties: f(0) = 7 , f(7)=0 , the orbit of 1 is {1,3,5} , and the point 6 is a fixed point. Explain carefully why f must have at least one period 7 point and at least one period 777 point. Write out the orbits of each of these points: point 5, point 6, and point 7. If you have difficulty drawing f to have all of the required properties, then draw f so as to have at least some of these properties (and indicate which ones you are including and excluding.)

Well, the orbit of 7 is evidently {0, 7}, because f takes 7 -> 0 -> 7.  And the orbit of 5 will be the same as the orbit of 1, since you are required to put them in the same orbit.  And the question demands that 6 be a fixed point, which means nothing more nor less than that the orbit of 6 be {6}.  So that's that part.

I would suggest that you make the function piecewise linear.

I hope this is some help.

As for the period 7 and period 777 points, you want Sharkovskii's theorem.  See  http://en.wikipedia.org/wiki/Sarkovskii%27s_theorem .

That really was helpful, thank you for the responses!

You are very welcome!