Just going through past papers as my anaysis exam is on Weds, but I'm quite stuck/unsure about 2 questions I've come across...
If g:[0,1] ---> [0,1] is a continuous function, show that there exists x in [0,1] such that g(x) = x.
Let F(x) = g(x) - x be defined on [0,1].
F is continuous on [0,1] since it is the difference of 2 continuous functions.
If F(0) = 0 or if F(1) =0 then we have our fixed point.
Otherwise, F(0) = g(0) - 0 > 0.
F(1) = g(1) - 1 < 0.
Since g(x) in [0,1] for all x in [0,1], by IVT there exists c in [0,1] such that F(c) = 0, hence g(c) = c.
Is this an OK answer, and would it be worthy of 8 marks?
Let f: R ---> R be a continuous function, such that f(x) in the rationals for all x. Show that f is a constant function.
This one I'm really stuck on and although it seems alot easier I have no idea what to say for it...can anyone please help?
Thanks in advance!