Suppose we have a set $M$, a $C^k$-atlas $\{\varphi_\alpha:U_\alpha\to\mathbb{R}^n\}_{\alpha\in A}$ on $M$, and a function $f:M\to\mathbb{R}$.
Is there any relation or restriction between the differentiability class of a transition map $\varphi_\alpha\circ\varphi_\beta^{-1}$ (class $C^k$) and that of $f\circ\varphi_\gamma^{-1}$ (class $C^l$) for any $\alpha, \beta, \gamma$ in $A$?
For instance, is $\varphi_\alpha\circ\varphi_\beta^{-1}$ in the differentiability class of $f\circ\varphi_\gamma^{-1}$ ($k \ge l$)? Since $f\circ\varphi_\beta^{-1}=f\circ(\varphi_\alpha^{-1}\circ\varphi_\alpha)\circ\varphi_\beta^{-1}=(f\circ\varphi_\alpha^{-1})\circ(\varphi_\alpha\circ\varphi_\beta^{-1})$, where $f\circ\varphi_\beta^{-1}$ and $f\circ\varphi_\alpha^{-1}$ are in class $C^l$ while $\varphi_\alpha\circ\varphi_\beta^{-1}$ is in class $C^k$.