Let \(X\subseteq\mathbb{R}^ n\) and \(Z\subseteq\mathbb{R}\) be nonempty sets endowed with a binary relation \(\succeq_ 1\) and \(\succeq_ 2\), respectively. Given a function \(f: X\to Z\), we say that \(f\) preserves \(\succeq_ 1\) with \(\succeq_ 2\) if \(f(z)\succeq_ 2 f(x)\) whenever \(x,y\in X\) satisfy \(y\succeq_ 1 x\). Under the assumption that \(f: X\to Z\) is continuously differentiable the authors state necessary as well as sufficient conditions assuring that \(f\) preserves \(\succeq_ 1\) with \(\succeq_ 2\). By specializing the relations \(\succeq_ 1\) and \(\succeq_ 2\), there are obtained different kinds of monotonicity which are linked with generalized convexity.