It is a theorem of ZF that every sequentially continuous function $\mathbb{R}\to\mathbb{R}$ is continuous. The proof is usually given in ZFC (and indeed, Choice is necessary to assert that sequential continuity at a point implies continuity at that point), but a proof can be given in ZF that sequential continuity everywhere implies continuity everywhere: see Herrlich, The Axiom of Choice (2006), theorem 3.15 and subsequent remarks on page 30.
(The proof in ZF is bizarre and somewhat counterintuitive, and since it only works for continuity everywhere, it seems quite defensible to use Choice to prove this.)