 Solution

Bounded means finite diameter; sequential compactness means every sequence has a convergent subsequence. A failure of uniform continuity supplies two close sequences whose image distances stay apart; a convergent subsequence contradicts continuity. Into a discrete metric, continuity is local constancy, and compactness upgrades the radii uniformly. The Cantor set is closed in compact $[0,1]$, hence sequentially compact. Uniform local constancy gives a positive separation scale, so finitely many initial Cantor-code (binary) digits determine $f$. Removing zero destroys compactness: the hinted function reading the digit after the first $1$ is continuous at every remaining sequence but depends on arbitrarily late digits.