Topology function from unit disk f(u) = λu?
Let Δ be the closed unit disk, let Γ = {z ∈ complex numbers : |z|=1}. Let f be a continuous function Δ → Γ. Show that for every λ ∈ Γ there is a u ∈ Γ with f(u) = λu.
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Since Γ is contained in Δ, f is also a continuous function from Γ to itself. Write g(u) = f(u)/u, where f is restricted to Γ, then like f, g is a continuous function of Γ to itself. All we need to prove now is that for any λ in Γ, there is a u in Γ with g(u) = λ.
Parametrize Γ using u = exp(2πi x) with x in the interval [0,1). Then h(x) = g(exp(2πi x)) is a continuous map from [0,1) to itself, and so by the intermediate value theorem its takes all values on [0,1).