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.)