if $f\in C(\mathbb{\overline{D}})$ and $\forall z\in\partial\mathbb{D}.f(z)=z$ then $\exists z\in\mathbb{D}.f(z)=0$

Question

if $f\in C(\mathbb{\overline{D}})$ and $\forall z\in\partial\mathbb{D}.f(z)=z$ then $\exists z\in\mathbb{D}.f(z)=0$

f is not analytical only Continuous .

solve with complex analysis tools

I thought looking at Winding numbers , but I reached a deadend.

Answer 1

Suppose $f$ never vanishes. Then $\gamma(s,t):=f(se^{it})$ is a homotopy equivalence between the constant curve $f(0)$ and the circle in $\mathbb{C}-\{0\}$. This, however, is impossible: the winding number around zero of the two curves is different and we know it is homotopically invariant.

(The fact that the winding number is homotopically invariant follows from Cauchy's integral theorem)