Let A and B be unital C ∗ -algebras acting on some Hilbert spaces. We investigate several topological properties of n-positive and n-monotone maps. It is shown that every 3-positive map Φ : (A , ∥ · ∥) → (B, SOT) is continuous, where SOT denotes the strong operator topology. Furthermore, we show that a 3-positive map is norm-continuous if it is norm-continuous at some positive invertible operator. Moreover, we prove that every 2-monotone map is norm-continuous. In addition, we show that in the definition of continuous Lieb function, the monotonicity condition is unnecessary. Finally, some interrelations between n-monotonicity and (n + 1)-positivity of positive nonlinear maps are presented. Several counterexamples illustrate the tightness of the results.

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