1) Let f be continuous at x₀ and
f(x+y) = f(x)+ f(y) for all x,y E R. Prove that f is continuous for all x E R.

2) Suppose f is a continuous surjective function from [a,b] to [a,b].Prove that there is at least one point x in [a,b] such that f(x) = x.

I think these problems will help me a lot in finishing the sheet.

Thanks
Cheers
Niranjan.

1) Fix any x in R. We will prove f is continuous at x. Well we have:

f(x') = f(x - x₀ ) + f((x' - x) + x₀ )

Letting x' -> x we have f((x' - x) + x₀ ) -> f(x₀ ) by continuity at x₀ and as f(x - x₀ ) is constant we get:

lim(x'-> x) f(x') = f(x - x₀ ) + f(x₀ ) = f(x)

so we're done. You can also do this with epsilons and deltas if that's the definition of "continuous" that you've been taught. By the way, can you see from here that the only solutions to the equation are f(x) = kx for some constant k?

2) Consider the function g(x) = f(x) - x. It suffices to show that g(x) has a root in [a,b]. To show this prove that there exists p,q with f(p) < = 0 and f(q) > = 0 then apply the intermediate value theorem.

The result holds for functions that aren't surjective, and the proof is simpler; just consider g(a) and g(b). One is < = 0, the other is > = 0. Hence g(x) = 0 for some x by the intermediate value theorem, and we're done.