Cardinality of T (holomorphic function)

Question

let D be a unit disk and T={f:D-C such that f is analytic on D and satisfies f(1/2n)=1/2n and f(1/2n+1)=1/2n for all n belongs to natural number } then what is the cardinality of set T . My try - my intention is to used the idenity theorem , if i define g(z)=f(z)-z then g(1/2n)=f(1/2n)-1/2n =0 for all n =1,2,3______ , since g(z) is also Analytic on D so analytic at 0 also , since 1/2n converges to 0 imply g(1/2n) converges to g(0) => g(0)=0 but zeroes of a non constant analytic function are isolated , so g(z) is identically zero on D so f(z)=z , also , f(1/2n+1)=1/2n but if we put on f(z)=z iam getting , f(1/2n+1)=1/2n+1 not 1/2n so T is empty set ??