Sufficient conditions for a norm-to-weak$^*$ continuous mapping $f: X \rightarrow X^*$ being monotone or submonotone are established by its Fréchet and normal coderivatives, where $X$ is an Asplund space with its dual space $X^*$. Under some additional assumptions, they are also necessary conditions. Among other things, we obtain a criterion for the monotonicity of continuous mappings which extends the following classical result: a differentiable mapping $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is monotone if and only if for each $x \in \mathbb{R}^n$ the Jacobian matrix $\nabla F(x)$ is positive semi-definite; see [22, Proposition 12.3]. As a by-product, sufficient conditions for a function being convex or approximately convex are given.