Originally Posted by

**hunkydory19** 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?

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