Let $M$ and $N$ be smooth manifolds with corresponding maximal atlases $A_M$ and $A_N$. We say that a map $f : M \to N$ is of class $C^r$ (or $r$-times continuously differentiable) at $p \in M$ if there exists a chart $(V, y)$ from $A_N$ with $f(p) \in V$, and a chart $(U, x)$ from $A_M$ with $p \in U$, such that $f (U) \subset V$ and $y\circ f \circ x^{-1}$ is of class $C^r$.
The above appears ambiguous to me. What if there are two other charts $(U',x')$ and $(V',y')$ with $p \in U'$ and $f(U') \subset V'$ but $y'\circ f \circ x'^{-1}$ not of class $C^r$? Will $f$ still be of class $C^r$ at $p$?
It may be the case that $y'\circ f\circ x'^{-1} : U' \to V'$ is not $C^r$, but its restriction to a sufficiently small open neighbourhood of $p$ will be $C^r$. To see this, note that (on an appropriate domain containing $p$)
\begin{align*} y'\circ f\circ x'^{-1} &= y'\circ \left(y^{-1}\circ y\right)\circ f\circ \left(x^{-1}\circ x\right)\circ x'^{-1}\\ &= \left(y'\circ y^{-1}\right)\circ \left(y\circ f\circ x^{-1}\right)\circ\left(x\circ x'^{-1}\right). \end{align*}
The maps $y'\circ y^{-1}$ and $x\circ x'^{-1}$ are transition maps for $N$ and $M$ respectively. As the atlases are smooth, $y'\circ y^{-1}$ and $x\circ x'^{-1}$ are smooth functions (and therefore $C^r$). As the composition of $C^r$ functions is a $C^r$ function, and $y\circ f\circ x^{-1}$ is $C^r$, $y'\circ f\circ x'^{-1}$ is $C^r$.