Proving equality of 2 continuous function on a point under certain conditions

Question

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Assume S(z) and S'(z) are two complex functions both continuous on X ={ z$\epsilon$ Complex numbers : z$\geq$1 } which are equal on |z|>1.

By which result it can be deduced that both S'(z) and S(z) are equal on |z|=1.

Answer 1

If $|z|=1$ then $z_n =(1+\frac 1 n )z \to z$ and $|z_n| >1$ for all $n$. Hence $S(z)=\lim S(z_n)=\lim S'(z_n)=S'(z)$.