Let \(k \ge 1\) be a natural number, \(L\) and \(M\) be \(k\) times differentiable manifolds with the \(C^k\)-atlases \((U_{i},\alpha _{i})\), \(i\in I\) and \((V_{j},\beta _{j})\), \(j\in J\) and let \(1\leq \ell \leq k\). A continuous function \[\varphi \colon L\longrightarrow M\,\] is called a \(C^{\ell }\)-differentiable function, if the functions \[\beta _{j}\circ \varphi \circ (\alpha _{i})^{-1}\colon \alpha _{i}(\varphi ^{-1}(V_{j})\cap U_{i})\longrightarrow V_{j}'\,\]
are \(l\) times continuously differentiable for all \(i\in I\) and all \(j\in J\).